\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
2006  International Conference in Honor of Jacqueline Fleckinger.
\newline \emph{Electronic Journal of Differential Equations},
Conference 16, 2007, pp. 35--58.
\newline ISSN: 1072-6691.
URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document} \setcounter{page}{35}

\title[\hfilneg EJDE/Conf/16 \hfil Ground-state compactness]
{Compactness for a Schr\"odinger operator\\
 in the ground-state space over $\mathbb{R}^N$}


\author[B. Alziary, P. Tak\'a\v{c} \hfil EJDE/Conf/16 \hfilneg]
{B\'en\'edicte Alziary,  Peter Tak\'a\v{c}} % in alphabetical order

\address{B\'en\'edicte Alziary \newline
CEREMATH -- UMR MIP,
Universit\'e Toulouse~1 (Sciences Sociales),
21 All\'ees de Brienne,
F--31000 Toulouse Cedex, France}
\email{alziary@univ-tlse1.fr}

\address{Peter Tak\'a\v{c} \newline
Institut f\"ur Mathematik,
Universit\"at Rostock,
Universit\"atsplatz 1,
D--18055 Rostock, Germany}
\email{peter.takac@uni-rostock.de}


\thanks{Published May 15, 2007.}
\subjclass[2000]{47A10, 35J10, 35P15, 81Q15}
\keywords{  Ground-state space; compact resolvent; Schr\"odinger operator;
\hfill\break\indent
  monotone radial potential; maximum and anti-maximum principle;
\hfill\break\indent
  comparison of ground states; asymptotic equivalence}

\dedicatory{Dedicated to  Jacqueline Fleckinger 
  on the occasion of \\ an international conference in her honor}

\begin{abstract}
 We investigate the compactness of the resolvent
 $(\mathcal{A} - \lambda I)^{-1}$ of the Schr\"odinger operator
 $\mathcal{A} = - \Delta + q(x)\bullet$
 acting on the Banach space $X$,
\begin{equation*}
  X = \{ f\in L^2(\mathbb{R}^N): f / \varphi\in L^\infty(\mathbb{R}^N) \} ,\quad
  \| f\|_X = \mathop{\rm ess\,sup}_{\mathbb{R}^N} (|f| / \varphi)\, ,
\end{equation*}
 $X\hookrightarrow L^2(\mathbb{R}^N)$, where $\varphi$ denotes
 the ground state for $\mathcal{A}$ acting on $L^2(\mathbb{R}^N)$.
 The potential
 $q: \mathbb{R}^N\to [q_0,\infty)$, bounded from below,
 is a ``relatively small'' perturbation of a radially symmetric potential
 which is assumed to be monotone increasing
 (in the radial variable) and growing somewhat faster than
 $|x|^2$ as $|x|\to \infty$.
 If $\Lambda$ is the ground state energy for $\mathcal{A}$, i.e.\
 $\mathcal{A}\varphi = \Lambda\varphi$,
 we show that the operator
 $(\mathcal{A} - \lambda I)^{-1} : X\to X$
 is not only bounded, but also compact for
 $\lambda\in (-\infty, \Lambda)$.
 In particular, the spectra of $\mathcal{A}$ in $L^2(\mathbb{R}^N)$ and $X$
 coincide; each eigenfunction of $\mathcal{A}$ belongs to $X$, i.e.,
 its absolute value is bounded by $\mathrm{const}\cdot \varphi$.
\end{abstract}

\maketitle

\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}


\section{Introduction} \label{s:Intro}

We investigate the compactness of the resolvent
$(\mathcal{A} - \lambda I)^{-1}$ of the Schr\"odinger operator
%
\begin{equation}
  \mathcal{A}\equiv \mathcal{A}_q\stackrel{{\rm def}}{=} - \Delta + q(x)\bullet
\label{op:Schroed_lam}
\end{equation}
%
acting not only on the standard Hilbert space $L^2(\mathbb{R}^N)$,
but also on the Banach space $X$,
%
\begin{equation}
  X\equiv X_q\stackrel{{\rm def}}{=}
  \{ f\in L^2(\mathbb{R}^N): f / \varphi\in L^\infty(\mathbb{R}^N) \} ,
\label{def:X}
\end{equation}
%
or on its predual space
$X^\odot = L^1(\mathbb{R}^N; \varphi \,\mathrm{d}x)$.
Here, $\varphi\equiv \varphi_q$ denotes the (normalized) ground state of
$\mathcal{A}$.
The electric potential $q(x)$ is assumed to be a continuous function
$q: \mathbb{R}^N\to \mathbb{R}$ such that
%
\begin{equation}
  q_0\stackrel{{\rm def}}{=} \inf_{\mathbb{R}^N} q > 0 \quad\mbox{and }\quad
  q(x)\to +\infty \mbox{ as } |x|\to \infty .
\label{cond:q_0}
\end{equation}
%
We denote by $\Lambda\equiv \Lambda_q$ the principal eigenvalue of
the operator $\mathcal{A}$
(also called the ground state energy); hence,
$\mathcal{A}\varphi = \Lambda\varphi$.
It follows from \eqref{cond:q_0} that
$(\mathcal{A} - \lambda I)^{-1}$ is a compact linear operator acting on
$L^2(\mathbb{R}^N)$ for every $\lambda\in (-\infty, \Lambda)$.
Furthermore,
$(\mathcal{A} - \lambda I)^{-1}$
is bounded as an operator acting on $X$,
by a standard argument using the weak maximum principle.
This operator, in general, is {\it not } compact on $X$;
for instance, not for $q(x) = |x|^2$ ($|x|\geq r_0 > 0$).

As a direct consequence of
the Riesz\--Schauder theory for compact linear operators
({Edwards} \cite[{\S}9.10, pp.\ 677--682]{Edwards} or
 {Yosida} \cite[Chapt.~X, Sect.~5, pp.\ 283--286]{Yosida}),
if $(\mathcal{A} - \lambda I)^{-1}$ happens to be compact on $X$
then, for instance,
all $L^2(\mathbb{R}^N)$\--eigenfunctions of $\mathcal{A}$
actually belong to $X$.
Moreover, one can prove an anti\--maximum principle as well.
We refer to
{Alziary}, {Fleckinger}, and {Tak\'a\v{c}}
\cite[Theorem 2.1, p.~128]{AFT-1} for $N=2$,
\cite[Theorem 2.1, p.~365]{AFT-2} for $N\geq 2$, and to
\cite[Sect.~3]{AFT-3}
for further details in applications of the compactness of
$(\mathcal{A} - \lambda I)^{-1}$ on~$X$.
The corresponding results, under different hypotheses, are stated in
Section~\ref{s:Main} below.

In the work reported in the present article we persue the search
(started in
 {Alziary}, {Fleckinger}, and {Tak\'a\v{c}} \cite{AFT-3})
for finding ``reasonable'' {\it sufficient conditions }
on the potential $q(x)$ that guarantee the compactness of
$(\mathcal{A} - \lambda I)^{-1}$ on $X$.
Different sufficient conditions on $q$, which also force
$(\mathcal{A} - \lambda I)^{-1}$ compact on $X$,
are formulated in \cite[Theorem 3.2(a)]{AFT-3}.
We will take advantage of some of the recent results obtained in
\cite{AFT-3} to treat potentials $q$ satisfying
%
\begin{equation}
\label{ineq:Q_1(r)<q<Q_2(r)}
  Q_1(|x|)\leq q(x)\leq Q_2(|x|) \quad\mbox{for all } x\in \mathbb{R}^N ,
\end{equation}
%
where
$Q_1, Q_2: \mathbb{R}_+\to (0,\infty)$
are some functions ($\mathbb{R}_+ = [0,\infty)$),
monotone increasing and continuous, such that
%
\begin{math}
  \int_1^{\infty} Q_1(r)^{-1/2} \,\mathrm{d}r < \infty
\end{math}
%
and the difference
$Q_2(r)^{1/2} - Q_1(r)^{1/2}$ is in some weighted Lebesgue space
$L^1( [1,\infty) ;\, w(r) \,{\rm d}r )$
with a suitable weight function
$w: [1,\infty)\to [1,\infty)$
which depends on the growth of $Q_2(r)$ as $r\to \infty$.
For instance, one may take $w(r)\equiv 1$ ($r\geq 1$) if
$Q_2(r)$ has at most power growth near infinity, i.e.,
$Q_2(r)\leq \mathrm{const}\cdot r^{\alpha}$
for $r\geq 1$, with some $\alpha\in (2,\infty)$; see
Lemma~\ref{lem-power-grow} in {\S}\ref{ss:power-grow} (the Appendix).


\begin{remark}\label{rem-ineq:Q_1(r)<q<Q_2(r)} \rm
If $q: \mathbb{R}^N\to \mathbb{R}$
is continuous and satisfies \eqref{cond:q_0},
then it is easy to see that both functions
%
\begin{equation}
\label{def:growth:q(x)}
  \tilde{Q}_1(r)\stackrel{{\rm def}}{=} \min_{|y|\geq r} q(y) \quad\mbox{and }\quad
  \tilde{Q}_2(r)\stackrel{{\rm def}}{=} \max_{|y|\leq r} q(y)
    \quad\mbox{of } r\in \mathbb{R}_+
\end{equation}
%
(where $y\in \mathbb{R}^N$)
are monotone increasing and continuous, together with
$\tilde{Q}_1(|x|)\leq q(x)\leq \tilde{Q}_2(|x|)$
for all $x\in \mathbb{R}^N$.
In particular, one may use the pair of functions
$(\tilde{Q}_1, \tilde{Q}_2)$ in place of $(Q_1,Q_2)$
in order to impose sufficient conditions on $q$ that force
$(\mathcal{A} - \lambda I)^{-1}$ compact on $X$.
%
\end{remark}


To be more specific, let us begin with
the radially symmetric eigenvalue problem
%
\begin{equation}
  \mathcal{A} v\equiv - \Delta v + q(|x|) v = \lambda v
  \quad\mbox{in }\, L^2(\mathbb{R}^N) ,\quad 0\neq v\in L^2(\mathbb{R}^N) ,
\label{ev:Schroed_lam}
\end{equation}
%
i.e., let $q(x)\equiv q(r)$ be radially symmetric, where $r = |x|\geq 0$.
First, consider the harmonic oscillator, that is,
$q(r) = r^2$ for $r\geq 0$.
One finds immediately that, except for the ground state
$\varphi$ itself, {\it no other } eigenfunction $v$ of $\mathcal{A}$
(associated with an eigenvalue $\lambda\neq \Lambda$)
can satisfy
$v / \varphi\in L^{\infty}(\mathbb{R}^N)$.
We refer to {Davies} \cite[Sect.\ 4.3, pp.\ 113--117]{Davies}
for greater details when $N=1$.
On the other hand, if
$q(r) = r^{2+\varepsilon}$ for $r\geq 0$ ($\varepsilon > 0$ -- a constant),
then
$v / \varphi\in L^{\infty}(\mathbb{R}^N)$
holds for {\it every } eigenfunction $v$ of $\mathcal{A}$,
again by results from {Davies} \cite{Davies},
Corollary 4.5.5 (p.~122) combined with Lemma 4.2.2 (p.~110) and
Theorem 4.2.3 (p.~111).
We refer to
{Davies} and {Simon} \cite[Theorem 6.3, p.~359]{DavSimon}
and
{M.~Hoffmann\--Ostenhof} \cite[Theorem 1.4(i), p.~67]{Hoffmann}
for the same result under much weaker restrictions on $q(x)$.
In our present article we impose similar restrictions.

Now assume $\lambda\in \mathbb{C}$, $f\in L^2(\mathbb{R}^N)$,
$f\geq 0$ and $f\not\equiv 0$ in~$\mathbb{R}^N$, and let
$u\in L^2(\mathbb{R}^N)$ be a solution of the equation
%
\begin{equation}
  - \Delta u + q(x) u = \lambda u + f(x)
  \quad\mbox{in }\, L^2(\mathbb{R}^N)
\label{e:Schroed_lam}
\end{equation}
%
in the sense of distributions on $\mathbb{R}^N$.
A related sufficient condition on $q$ is imposed in
{Alziary} and {Tak\'a\v{c}}
\cite[Theorem 2.1, p.~284]{AlziaryTak} to obtain
%
\begin{equation}
  u\geq c\varphi\quad\mbox{in $\mathbb{R}^N$ ({\it $\varphi$-positivity }) }
\label{ineq:phi_+}
\end{equation}
%
for $\lambda < \Lambda$, with some constant $c>0$.
Somewhat stronger sufficient conditions on $q$ and $f$ guarantee also
%
\begin{equation}
  u\leq - c\varphi\quad\mbox{in $\mathbb{R}^N$ ({\it $\varphi$-negativity }) }
\label{ineq:phi_-}
\end{equation}
%
provided $\Lambda < \lambda < \Lambda + \delta$, where
$\delta\equiv \delta(f) > 0$ is sufficiently small, $c>0$; see
{Alziary}, {Fleckinger}, and {Tak\'a\v{c}}
\cite[Theorem 2.1, p.~128]{AFT-1} for $N=2$ and
\cite[Theorem 2.1, p.~365]{AFT-2} for $N\geq 2$.
Moreover, in both inequalities
\eqref{ineq:phi_+} and \eqref{ineq:phi_-}
we have $c\equiv c(\lambda)\to +\infty$ as $\lambda\to \Lambda$.
Again, the harmonic oscillator $q(x)\equiv |x|^2$ and
a suitably chosen positive function $f$ provide
easy counterexamples to both,
\eqref{ineq:phi_+} and \eqref{ineq:phi_-}.

 From the proof of Theorem \ref{thm-X-compact}, Part~(a),
we will derive \eqref{ineq:phi_+} whenever
$f\in X^\odot$, $0\leq f\not\equiv 0$ in $\mathbb{R}^N$, and
$\lambda < \Lambda$.
This result is stated as Theorem \ref{thm-Positive-q(x)}.
Directly from Theorem \ref{thm-X-compact}, Part~(c),
we will derive also \eqref{ineq:phi_-} whenever $f\in X$,
$\int_{\mathbb{R}^N} f\varphi \,{\rm d}x > 0$, and
$\Lambda < \lambda < \Lambda + \delta$
($\delta > 0$ -- small enough).
This is the anti\--maximum principle in Theorem \ref{thm-Antimax-q(x)}.

The proof of Theorem \ref{thm-X-compact}
is based on the asymptotic equivalence of
$\varphi_{Q_1}(|x|)$ and $\varphi_{Q_2}(|x|)$ as $|x|\to \infty$,
for $Q_1$ and $Q_2$ as described above.
An important ingredient here is Lemma \ref{lem-Titchmarsh}
which generalizes
{Titchmarsh}' lemma \cite[Sect.~8.2, p.~165]{Titchmarsh}
applied in
{Alziary} and {Tak\'a\v{c}}
\cite[Lemma 3.2, p.~286]{AlziaryTak},
and in
{Alziary}, {Fleckinger}, and {Tak\'a\v{c}}
%\cite[Lemma 3.2, p.~132]{AFT-1} and \cite[Lemma 3.2, p.~366]{AFT-2}
\cite[p.~132]{AFT-1} and \cite[p.~366]{AFT-2},
with a slightly different class of potentials $Q_1(r)$ and $Q_2(r)$.
In {Alziary}, {Fleckinger}, and {Tak\'a\v{c}}
\cite[Lemma 4.1]{AFT-3}
the authors use a WKB\--type asymptotic formula due to
{Hartman} and {Wintner} \cite[eq.\ (xxv), p.~49]{Hartman-Wint}.

Asymptotic estimates for radially symmetric solutions of
the Schr\"odinger equation with $q(x) = Q_j(|x|)$ ($j=1,2$)
are combined with standard comparison results for solutions
with different, but pointwise ordered (nonradial) potentials
in order to control the asymptotic behavior of these solutions
at infinity, and thus retain the compactness of the resolvent
from the radially symmetric case
({\rm Proposition~\ref{prop-q_1<q_2}}).
In our approach it is crucial that the ground states
$\varphi_j(x)\equiv \varphi_{Q_j}(|x|)$
corresponding to the potentials $Q_j(|x|)$ ($j=1,2$)
are \emph{comparable }, that is,
$\varphi_1 / \varphi_2\in L^{\infty}(\mathbb{R}^N)$
(by Proposition~\ref{prop-Positive-Q(r)})
and
$\varphi_2 / \varphi_1\in L^{\infty}(\mathbb{R}^N)$
(by Corollary~\ref{cor-q_1<q_2}).

This article is organized as follows.
In the next section (Section~\ref{s:Hypotheses})
we describe the type of potentials $q(x)$ we are concerned with,
together with some basic notations.
Section~\ref{s:Main} contains our main results.
These results are proved in
Sections \ref{s:Prelim} through~\ref{s:Proofs}.

\section{Hypotheses and notation}
\label{s:Hypotheses}

We consider the Schr\"odinger equation \eqref{e:Schroed_lam}, i.e.,
%
\begin{equation*}
  - \Delta u + q(x) u = \lambda u + f(x)
  \quad\mbox{in }\, L^2(\mathbb{R}^N) .
%\label{e:Schroed_lam}
\end{equation*}
%
Here, $f\in L^2(\mathbb{R}^N)$ is a given function,
$\lambda\in \mathbb{C}$ is a complex parameter, and the potential
$q: \mathbb{R}^N\to \mathbb{R}$ is a continuous function;
we always assume that $q$ satisfies \eqref{cond:q_0}, i.e.,
%
\begin{equation*}
  q_0\stackrel{{\rm def}}{=} \inf_{\mathbb{R}^N} q > 0 \quad\mbox{and}\quad
  q(x)\to +\infty \quad \mbox{as } |x|\to \infty .
%\label{cond:q_0}
\end{equation*}
%
We interpret equation \eqref{e:Schroed_lam}
as the operator equation
$\mathcal{A} u = \lambda u + f$ in $L^2(\mathbb{R}^N)$, where
the Schr\"odinger operator \eqref{op:Schroed_lam},
%
\begin{equation*}
  \mathcal{A}\equiv \mathcal{A}_q\stackrel{{\rm def}}{=} - \Delta + q(x)\bullet
  \quad\mbox{on }\, L^2(\mathbb{R}^N) ,
%\label{op:Schroed_lam}
\end{equation*}
%
is defined formally as follows:
We first define the quadratic (Hermitian) form
%
\begin{equation}
  \mathcal{Q}_q(v,w)\stackrel{{\rm def}}{=} \int_{\mathbb{R}^N}
  \left(
    \nabla v\cdot \nabla\bar{w} + q(x) v\bar{w}
  \right)\,{\rm d}x
\label{e:q-form}
\end{equation}
%
for every pair $v,w\in \mathcal{V}_q$ where
%
\begin{equation}
  \mathcal{V}_q\stackrel{{\rm def}}{=}
  \{ f\in L^2(\mathbb{R}^N): \mathcal{Q}_q(f,f) < \infty \} .
\label{e:q-space}
\end{equation}
%
Then $\mathcal{A}$ is defined to be
the Friedrichs representation of the quadratic form
$\mathcal{Q}_q$ in $L^2(\Omega)$;
$L^2(\Omega)$ is endowed with the natural inner product
%
\[
 (v,w)_{ L^2(\mathbb{R}^N) }\stackrel{{\rm def}}{=} \int_{\mathbb{R}^N} v\bar{w} \,{\rm d}x ,\quad
  v,w\in L^2(\Omega) .
\]
%
This means that $\mathcal{A}$ is
a positive definite, selfadjoint linear operator on
$L^2(\Omega)$ with domain $\mathop{\rm dom}(\mathcal{A})$
dense in $\mathcal{V}_q$ and
%
\[
   \int_{\mathbb{R}^N} (\mathcal{A} v)\bar{w} \,{\rm d}x
  = \mathcal{Q}_q(v,w)
  \quad\mbox{for all } v,w\in \mathop{\rm dom}(\mathcal{A}) ;
\]
%
see {Kato} \cite[Theorem VI.2.1, p.~322]{Kato}.
Notice that $\mathcal{V}_q$ is a Hilbert space with the inner product
$(v,w)_q = \mathcal{Q}_q(v,w)$
and the norm
$\| v\|_{\mathcal{V}_q} = ((v,v)_q)^{1/2}$.
The embedding
$\mathcal{V}_q\hookrightarrow L^2(\mathbb{R}^N)$
is compact, by \eqref{cond:q_0}.

The principal eigenvalue $\Lambda\equiv \Lambda_q$ of the operator
$\mathcal{A}\equiv \mathcal{A}_q$
can be obtained from the Rayleigh quotient
%
\begin{equation}
  \Lambda\equiv \Lambda_q = \inf \left\{
    \mathcal{Q}_q(f,f) :
    f\in \mathcal{V}_q\ \mbox{ with }\ \| f\|_{L^2(\mathbb{R}^N)} = 1
    \right\} ,
  \quad \Lambda > 0 .
\label{e:Lambda}
\end{equation}
%
This eigenvalue is simple with the associated eigenfunction
$\varphi\equiv \varphi_q$ normalized by
$\varphi > 0$ throughout $\mathbb{R}^N$ and
$\|\varphi\|_{L^2(\mathbb{R}^N)} = 1$;
$\varphi$ is a minimizer for the Rayleigh quotient above.
The reader is referred to
{Edmunds} and {Evans} \cite{EdmEvans}
or
{Reed} and {Simon} \cite[Chapt.\ XIII]{RSimon-IV}
for these and other basic facts about Schr\"odinger operators.

We set $r = |x|$ for $x\in \mathbb{R}^N$, so $r\in \mathbb{R}_+$, where
$\mathbb{R}_+\stackrel{{\rm def}}{=} [0,\infty)$.
If $q$ is a radially symmetric potential,
$q(x) = q(r)$ for $x\in \mathbb{R}^N$,
then also the eigenfunction $\varphi$ must be radially symmetric.
This follows directly from $\Lambda$ being a simple eigenvalue.

Since our technique is based on a perturbation argument for
a relatively small perturbation of a radially symmetric potential,
which is assumed to satisfy
certain differentiability and growth conditions
in the radial variable $r = |x|$, $r\in \mathbb{R}_+$,
we bound the potential $q: \mathbb{R}^N\to \mathbb{R}$
by such radially symmetric potentials from below and above.

In order to formulate our hypotheses on the potential
$q(x)$, $x\in \mathbb{R}^N$, we first introduce
the following class \eqref{class:Q(r)} of {\it auxiliary functions }
$Q(r)$ of $r=|x|\geq 0$:


\begin{enumerate}
%
\item[(Q)]
\makeatletter
\def\@currentlabel{Q}\label{class:Q(r)}
\makeatother
%
$Q: \mathbb{R}_+\to (0,\infty)$ is a continuous function
that is monotone increasing in some interval
$[r_0,\infty)$, $0 < r_0 < \infty$, and satisfies
%
\begin{equation}
 \int_{r_0}^\infty Q(r)^{-1/2} \,\mathrm{d}r < \infty .
\label{e:int-Q^-1/2}
\end{equation}
%
\end{enumerate}


In particular, we have $Q'(r)\geq 0$ for a.e.\ $r\geq r_0$, and
$Q(r)\to \infty$ as $r\to \infty$.


Example \ref{exam-q(r):mono} in the Appendix, {\S}\ref{ss:Examples},
is essential for understanding potentials of class \eqref{class:Q(r)}.
The potential $q(x) = q(|x|)$ exhibited in this example belongs to
class \eqref{class:Q(r)}, but does {\it not } belong to
the analogue of this class defined in
{Alziary}, {Fleckinger}, and {Tak\'a\v{c}} \cite{AFT-3}.
There, instead of $Q$ monotone increasing in $[r_0,\infty)$,
it is required that
there be a constant $\gamma$, $1 < \gamma\leq 2$, such that
%
\[
 \int_{r_0}^\infty
  \Big|    \frac{{\rm d}}{{\rm d}r} \Big( Q(r)^{-1/2} \Big)
  \Big|^{\gamma} Q(r)^{1/2} \,{\rm d}r < \infty .
\label{e:int-deriv}
\]
%
As a consequence, in our present paper we do not need to employ
the Hartman\--Wintner asymptotic formula from \cite{Hartman-Wint},
eq.\ (xxv) on p.~49 or eq.\ (158) on p.~80.
Condition \eqref{e:int-deriv} originally appeared in the work of
{Hartman} and {Wintner} \cite{Hartman-Wint},
on p.~49, eq.\ (xxiv), and on p.~80, eq.\ (157).

We impose the following hypothesis on the growth of $q$ from below:

\begin{enumerate}
%
\item[(H1)]
\makeatletter
\def\@currentlabel{H1}\label{hyp:growth:q(r)}
\makeatother
%
There exists a function
$Q: \mathbb{R}_+\to (0,\infty)$ of class {\rm (Q)}
such that
%
\begin{equation}
  q(x) - \Lambda + \frac{(N-1)(N-3)}{4 r^2}
  \geq Q(r) > 0
    \quad\mbox{holds for all } |x| = r > r_0 .
\label{growth:q(r)}
\end{equation}
%
\end{enumerate}



\begin{remark}\label{rem-growth:q(r)} \rm
The term $- \Lambda + \frac{(N-1)(N-3)}{4 r^2}$
has been added for convenience only; it may be left out by replacing
$Q(r)$ by $Q(r) + \Lambda + 1$ if $N\geq 1$
and taking also $r_0 > 0$ large enough if $N=2$.
\end{remark}

In several results we need stronger hypotheses than
\eqref{hyp:growth:q(r)}.
Writing $x = r x'$
($x\in \mathbb{R}^N\setminus \{\mathbf{0}\}$)
with the radial and azimuthal variables
$r = |x|$ and $x'= x/|x|$, respectively,
we frequently impose the following stronger restrictions on
the growth of $q(x)$ in $r$ and the variation of $q(x)$ in $x'$:

We assume that
%
\begin{enumerate}
%
\item[(H2)]
\makeatletter
\def\@currentlabel{H2}\label{hyp:growth:q(x)}
\makeatother
%
There exist two functions
$Q_1, Q_2: \mathbb{R}_+\to (0,\infty)$ of class \eqref{class:Q(r)}
and two positive constants
$C_{12}, r_0\in (0,\infty)$, such that
%
\begin{equation}
\label{growth:q(x)}
  Q_1(|x|)\leq q(x)\leq Q_2(|x|)\leq C_{12}\, Q_1(|x|)
    \quad\mbox{for all } x\in \mathbb{R}^N \,,
\end{equation}
%
together with
%
\begin{equation}
\label{e:int-diff}
  \int_{r_0}^\infty ( Q_2(s) - Q_1(s) ) \int_{r_0}^s \exp
    \Big( - \int_r^s [ Q_1(t)^{1/2} + Q_2(t)^{1/2} ] \,\mathrm{d}t
    \Big)
  \,\mathrm{d}r \,\mathrm{d}s
  < \infty \,.
\end{equation}
%
\end{enumerate}


In fact, it suffices to assume inequalities \eqref{growth:q(x)}
only for all $|x| = r > r_0$ with $r_0 > 0$ large enough, provided
$q: \mathbb{R}^N\to (0,\infty)$ is a continuous function.
Indeed, then one can find some extensions
$\tilde{Q}_1, \tilde{Q}_2: \mathbb{R}_+\to (0,\infty)$
of class \eqref{class:Q(r)} of (the restrictions of) functions
$Q_1, Q_2: [r_0 + 1, \infty)\to (0,\infty)$, respectively,
from $[r_0 + 1, \infty)$ to $\mathbb{R}_+$ such that
$\tilde{Q}_j(r) = Q_j(r)$ for $r\geq r_0 + 1$; $j=1,2$, and
inequalities \eqref{growth:q(x)} hold for all $x\in \mathbb{R}^N$
with $\tilde{Q}_j$ in place of $Q_j$.

\begin{remark}\label{rem-int-diff} \rm
(a)
Assuming \eqref{growth:q(x)}, we will show in
the Appendix, {\S}\ref{ss:power-grow},
that the latter condition, \eqref{e:int-diff},
is satisfied in the following two cases:
%
\begin{itemize}
%
\item[{\rm (i)}]
when $Q_2$ has at most power growth near infinity and the condition
%
\begin{equation}
 \int_{r_0}^\infty
    \left( Q_2(r)^{1/2} - Q_1(r)^{1/2} \right) \,{\rm d}r < \infty
    \quad\mbox{for some } 0 < r_0 < \infty
\label{equiv:int-diff}
\end{equation}
%
is valid; or
%
\item[{\rm (ii)}]
when $Q_2$ has at most exponential power growth near infinity,
i.e.,
$Q_2(r)\leq$\break $\gamma\cdot \exp( \beta r^{\alpha} )$
for all $r\geq r_0$, where
$\alpha, \beta, \gamma > 0$ and $r_0 > 0$ are some constants, and
%
\begin{equation}
 \int_{r_0}^\infty
    \left( Q_2(r)^{1/2} - Q_1(r)^{1/2} \right)
       \left[ 1 + \log^+ ( Q_1(r)^{1/2} + Q_2(r)^{1/2} ) \right]
    \,\mathrm{d}r < \infty \,.
\label{exp:int-diff}
\end{equation}
%
\end{itemize}
%
Clearly, condition \eqref{exp:int-diff} is stronger than
\eqref{equiv:int-diff}.

{\rm (b)}
With regard to Remark~{\rm \ref{rem-growth:q(r)}}
and from the point of view of spectral theory, the case
%
\begin{equation}
\label{growth:0<Q_2-Q_1<C}
  0\leq Q_2(r) - Q_1(r)\leq C \quad\mbox{for all } r\geq r_0
\end{equation}
%
is of importance.
Here, $C, r_0\in (0,\infty)$ are some constants.
Then the condition
(cf.\ \eqref{e:int-Q^-1/2})
%
\[
 \int_{r_0}^\infty Q_2(r)^{-1/2} \,\mathrm{d}r < \infty
%\label{e:int-Q_2^-1/2}
\]
%
implies also condition \eqref{e:int-diff}.
Indeed, combining \eqref{growth:0<Q_2-Q_1<C} with the fact that
$Q_1$ and $Q_2$ are of class \eqref{class:Q(r)}, we compute
%
\begin{equation}
\label{case:int-diff}
\begin{aligned}
& \int_{r_0}^\infty ( Q_2(s) - Q_1(s) ) \int_{r_0}^s \exp
    \Big( - \int_r^s [ Q_1(t)^{1/2} + Q_2(t)^{1/2} ] \,\mathrm{d}t
    \Big)
  \,\mathrm{d}r \,\mathrm{d}s
\\
& \leq C\int_{r_0}^\infty \int_{r_0}^s \exp
    \Big( - \int_r^s [ Q_1(r)^{1/2} + Q_2(r)^{1/2} ] \,\mathrm{d}t
    \Big)
    \,\mathrm{d}r \,\mathrm{d}s
\\
& = C\int_{r_0}^\infty \int_r^\infty
      \exp\Big( - [ Q_1(r)^{1/2} + Q_2(r)^{1/2} ] (s-r) \Big)
    \,\mathrm{d}s \,\mathrm{d}r
\\
& = C\int_{r_0}^\infty [ Q_1(r)^{1/2} + Q_2(r)^{1/2} ]^{-1}
    \,\mathrm{d}r
  < \infty \,.
\end{aligned}
\end{equation}
%
\end{remark}

We will see in {\S}\ref{ss:Compact-q(x)}
(the proof of Theorem~\ref{thm-X-compact}, {\rm Part~(a)})
that  \eqref{hyp:growth:q(x)}
guarantees that all ground states
$\varphi_{q}$, $\varphi_{Q_1}$, and $\varphi_{Q_2}$
are \emph{comparable }:
There exist some constants
$0 < \gamma_1\leq \gamma_2 < \infty$ such that
%
\begin{math}
       \gamma_1\varphi_{q} \leq \varphi_{Q_j}
  \leq \gamma_2\varphi_{q}
\end{math}
%
in $\mathbb{R}^N$; $j=1,2$.

\begin{remark}\label{rem-growth:q(x)}\rm
Define the functions
$\tilde{Q}_1, \tilde{Q}_2: \mathbb{R}_+\to (0,\infty)$
as in \eqref{def:growth:q(x)}.
Recall that both
$\tilde{Q}_1$ and $\tilde{Q}_2$
are monotone increasing and continuous, and satisfy
$\tilde{Q}_1(|x|)\leq q(x)\leq \tilde{Q}_2(|x|)$
for all $x\in \mathbb{R}^N$.
The functions
$Q_1, Q_2: \mathbb{R}_+\to (0,\infty)$
in condition \eqref{growth:q(x)}
being monotone increasing and continuous, as well,
this condition is equivalent to
%
\begin{equation}
\label{eq:growth:q(x)}
  Q_1(r)\leq \tilde{Q}_1(r)\leq \tilde{Q}_2(r)\leq Q_2(r)
  \leq C_{12}\, Q_1(r)
    \quad\mbox{for all } r\in \mathbb{R}_+ \,.
\end{equation}
%
As far as condition \eqref{e:int-diff} is concerned,
it holds equivalently for the pair
$(\tilde{Q}_1, \tilde{Q}_2)$ in place of $(Q_1,Q_2)$ provided
either of cases {\rm (i)} or {\rm (ii)} from Remark~\ref{rem-int-diff}
occurs.
%
\end{remark}

\section{Main results} \label{s:Main}

For any complex number $\lambda\in \mathbb{C}$ that is not an eigenvalue of
the operator
$\mathcal{A} = - \Delta + q(x)\bullet$
on $L^2(\mathbb{R}^N)$, we denote by
\[
    (\mathcal{A} - \lambda I)^{-1}
  = (- \Delta + q(x)\bullet - \lambda I)^{-1}
\]
the resolvent of $\mathcal{A}$ on $L^2(\mathbb{R}^N)$ given by
(cf.\ eq.~\eqref{e:Schroed_lam})
\[
  u(x) = \left[ (\mathcal{A} - \lambda I)^{-1} f \right](x) ,
  \quad x\in \mathbb{R}^N .
\]

Now let us fix any real number $\lambda < \Lambda$ and consider
the resolvent
$(\mathcal{A} - \lambda I)^{-1}$ on $L^2(\mathbb{R}^N)$.
By the weak maximum principle
(see the proof of Proposition~\ref{prop-Positive-Q(r)}),
the operator $(\mathcal{A} - \lambda I)^{-1}$
is \emph{positive}, that is, for
$f\in L^2(\mathbb{R}^N)$ and
$u = (\mathcal{A} - \lambda I)^{-1} f$ we have
%
\begin{equation}
  f\geq 0 \;\mbox{ a.e.\ in } \mathbb{R}^N \quad\Longrightarrow\quad
  u\geq 0 \;\mbox{ a.e.\ in } \mathbb{R}^N .
\label{e:max-princ_L^2}
\end{equation}
%
Consequently, given any constant $C>0$, we have also
%
\begin{equation}
  |f|\leq C\varphi \;\mbox{ in } \mathbb{R}^N \quad\Longrightarrow\quad
  |u|\leq C (\Lambda - \lambda)^{-1} \varphi \;\mbox{ in } \mathbb{R}^N ,
\label{e:max-princ_X}
\end{equation}
%
by linearity.
We denote by
$\mathcal{K}\vert_{X}$ the restriction of
$\mathcal{K} = (\mathcal{A} - \lambda I)^{-1}$
to the Banach space $X$ defined in \eqref{def:X}.
Hence,
$\mathcal{K}\vert_{X}$ is a bounded linear operator on $X$
with the operator norm $\leq (\Lambda - \lambda)^{-1}$, by
\eqref{e:max-princ_X}.

Clearly, $X$ is the dual space of the Lebesgue space
$X^\odot = L^1(\mathbb{R}^N; \varphi \,{\rm d}x)$
with respect to the duality induced by the natural inner product
on $L^2(\mathbb{R}^N)$.
The embeddings
\[
  X\hookrightarrow L^2(\mathbb{R}^N)\hookrightarrow X^\odot
\]
are dense and continuous.
Furthermore, $\mathcal{K}$ possesses a unique extension
$\mathcal{K}\vert_{X^\odot}$
to a bounded linear operator on $X^\odot$
(by Lemma \ref{lem-Riesz-Thorin} below).
Finally, it is obvious that
$\mathcal{K}\vert_{X} : X\to X$
is the adjoint of
$\mathcal{K}\vert_{X^\odot} : X^\odot\to X^\odot$.

\subsection{Main theorems}
\label{ss:Main-thm}

Throughout this paragraph we assume that $q(x)$ is a potential
that satisfies ~\eqref{hyp:growth:q(x)}.
Under this hypothesis we are able to show
the following \emph{ground\--state positivity } of
the weak solution to the Schr\"odinger equation
\eqref{e:Schroed_lam} in $X^\odot$.


\begin{theorem}\label{thm-Positive-q(x)}
Let  \eqref{hyp:growth:q(x)} be satisfied and let
$-\infty < \lambda < \Lambda$.
Assume that  $f\in X^\odot$ satisfies
$f\geq 0$ almost everywhere and
$f\not\equiv 0$ in~$\mathbb{R}^N$.
Then the (unique) solution $u\in X^\odot$ to
equation \eqref{e:Schroed_lam}
(in the sense of distributions on $\mathbb{R}^N$)
is given by
$u = (\mathcal{A} - \lambda I)^{-1}\vert_{X^\odot} f$
and satisfies
$u\geq c\varphi$ almost everywhere in $\mathbb{R}^N$,
with some constant  $c\equiv c(f) > 0$.
%
\end{theorem}

In the literature, the inequality $u\geq c\varphi$
is often called briefly \emph{$\varphi$-positivity}.
In {Protter} and {Weinberger}
\cite[Chapt.~2, Theorem~10, p.~73]{ProtterWein},
a similar result is referred to as
the \emph{generalized maximum principle}.

This result has been established in
{Alziary} and {Tak\'a\v{c}}
\cite[Theorem 2.1, p.~284]{AlziaryTak}
under slightly different growth hypotheses on
the potentials $Q_1$ and $Q_2$
in conditions \eqref{growth:q(x)} and \eqref{e:int-diff}.
In \cite[eq.~(2), p.~283]{AlziaryTak},
a closely related class~\eqref{class:Q(r)} is used where $Q(r)$
still satisfies a condition similar to \eqref{e:int-Q^-1/2},
namely,
%
\begin{math}
  \int_{r_0}^\infty Q(r)^{-\beta} \,{\rm d}r < \infty
%\label{e:int-Q^-1/2}
\end{math}
%
with some constant $\beta\in (0,1/2)$.
Also the ``potential variation'' condition \eqref{e:int-diff}
in our present work is somewhat different from that assumed in
\cite[eq.~(5), p.~283]{AlziaryTak}.
%
Nevertheless, our proof of Theorem~\ref{thm-Positive-q(x)}
follows similar steps as does
the proof of Theorem 2.1 in \cite[pp.\ 289--290]{AlziaryTak}.
Theorem~\ref{thm-Positive-q(x)} will be proved in
Section~\ref{s:Proofs}, first for
$q(x) = Q(|x|)$ of class~\eqref{class:Q(r)},
as Proposition~\ref{prop-Positive-Q(r)}
in {\S}\ref{ss:Positive-Q(r)},
and then in its full generality in {\S}\ref{ss:Positive-q(x)},
after the proof of Theorem~\ref{thm-X-compact},
a part of which will be needed
(stated below as Corollary~\ref{cor-X-compact}).

The central result of this paper is the following compactness theorem.
Indeed, it provides answers to some questions about the solution $u$
of problem \eqref{e:Schroed_lam}, such as
$u\in X$ and its $\varphi$-positivity or $\varphi$-negativity.
Moreover, it also guarantees that the spectrum of the operator
$\mathcal{A} = - \Delta + q(x)\bullet$ is the same in each of the spaces
$L^2(\mathbb{R}^N)$, $X$, and $X^\odot$.

\begin{theorem}\label{thm-X-compact}
Let \eqref{hyp:growth:q(x)} be satisfied.
Then we have the following three statements for the resolvent
$\mathcal{K} = (\mathcal{A} - \lambda I)^{-1}$ of
$\mathcal{A}$ on $L^2(\mathbb{R}^N)$:
%
\begin{enumerate}
%
\item[{\rm (a)}]
%
If  $-\infty < \lambda < \Lambda$
then both operators  $\mathcal{K}\vert_{X}: X\to X$ and
$\mathcal{K}\vert_{X^\odot}: X^\odot\to X^\odot$
are compact  (and positive, see \eqref{e:max-princ_L^2}).
%
\item[{\rm (b)}]
%
If  $\lambda\in \mathbb{C}$ is an eigenvalue of $\mathcal{A}$, that is,
$\mathcal{A} v = \lambda v$ for some $v\in L^2(\mathbb{R}^N)$, $v\neq 0$,
then $v\in X$ (${}\subset L^2(\mathbb{R}^N)\subset X^\odot$) and
$\lambda\in \mathbb{R}$, $\lambda\geq \Lambda$.
%
\item[{\rm (c)}]
%
If  $\lambda\in \mathbb{C}$ is  \emph{not } an eigenvalue of  $\mathcal{A}$,
then the restriction $\mathcal{K}\vert_{X}$ of
$\mathcal{K}$ to $X$ is a bounded linear operator from $X$ into itself
and, moreover,
$\mathcal{K}$ possesses a unique extension
$\mathcal{K}\vert_{X^\odot}$
to a bounded linear operator from $X^\odot$ into itself.
Again, both operators  $\mathcal{K}\vert_{X}: X\to X$ and
$\mathcal{K}\vert_{X^\odot}: X^\odot\to X^\odot$
are compact.
%
\end{enumerate}
%
\end{theorem}

Part~(a) is the most difficult one to prove.
Since $\mathcal{K}\vert_{X}: X\to X$ is compact
{\it if and only if }
$\mathcal{K}\vert_{X^\odot}: X^\odot\to X^\odot$
is compact, by Schauder's theorem
({Edwards} \cite[Corollary 9.2.3, p.~621]{Edwards} or
 {Yosida} \cite[Chapt.~X, Sect.~4, p.~282]{Yosida}),
it suffices to prove that either of them is compact.
Thus, our proof of Part~(a) begins with
the compactness of the restriction of $\mathcal{K}\vert_{X}$ to
(the corresponding subspace of)
radially symmetric functions with
$q(x) = Q(|x|)$ of class \eqref{class:Q(r)}
and only for $\lambda < \Lambda$,
see Lemma \ref{lem-X-compact}.
So we may apply Schauder's theorem to get the compactness of
the restriction of
$\mathcal{K}\vert_{X^\odot}$ to
radially symmetric functions with $q=Q$.
Then we extend this result to
$\mathcal{K}\vert_{X^\odot}$ on $X^\odot$ with $q=Q$ again,
see Proposition \ref{prop-X-compact}.
Finally, from there we derive that
$\mathcal{K}\vert_{X^\odot}$ is compact for any $q(x)$ satifying
 \eqref{hyp:growth:q(x)},
first only for $\lambda < \Lambda$ and then for any $\lambda\in \mathbb{C}$
that is not an eigenvalue of $\mathcal{A}$,
see Section~\ref{s:Proofs}, {\S}\ref{ss:Compact-q(x)}.
Parts (b) and~(c) are proved immediately thereafter;
they will be derived from Part~(a) by standard arguments based on
the Riesz\--Schauder theory of compact linear operators
({Edwards} \cite[{\S}9.10, pp.\ 677--682]{Edwards} or
 {Yosida} \cite[Chapt.~X, Sect.~5, pp.\ 283--286]{Yosida}).

As a direct consequence of Theorem \ref{thm-X-compact}, Part~(a), the
following corollary establishes the equivalence of the ground states.

\begin{corollary}\label{cor-X-compact}
Let \eqref{hyp:growth:q(x)} be satisfied.
Then the ground states
$\varphi_{q}$, $\varphi_{Q_1}$, and  $\varphi_{Q_2}$
corresponding to the potentials $q$, $Q_1$, and  $Q_2$, respectively,
are comparable, that is,
there exist some constants
$0 < \gamma_1\leq \gamma_2 < \infty$ such that
%
\begin{math}
       \gamma_1\varphi_{q} \leq \varphi_{Q_j}
  \leq \gamma_2\varphi_{q}
\end{math}
%
in $\mathbb{R}^N$; $j=1,2$.
Equivalently, we have $X_{q} = X_{Q_1} = X_{Q_2}$.
%
\end{corollary}

A classical use of the compactness result,
Theorem \ref{thm-X-compact}, Part~(c), is
the {\it anti\--maximum principle } for the Schr\"odinger operator
$\mathcal{A} = - \Delta + q(x)\bullet$
which complements the ground\--state positivity of
Theorem~\ref{thm-Positive-q(x)}.

\begin{theorem}\label{thm-Antimax-q(x)}
Let  \eqref{hyp:growth:q(x)} be satisfied and let
$f\in X$ satisfy
$\int_{\mathbb{R}^N} f\varphi \,{\rm d}x$ $> 0$.
Then there exists a number  $\delta\equiv \delta(f) > 0$
such that, for every
$\lambda\in (\Lambda, \Lambda + \delta)$,
the inequality $u\leq - c\varphi$ is valid a.e.\ in $\mathbb{R}^N$
with some constant  $c\equiv c(f) > 0$.
%
\end{theorem}


This important consequence of Theorem \ref{thm-X-compact}
was presented also in {Alziary}, {Flec\-kinger}, and {Tak\'a\v{c}}
\cite[Theorem 3.4]{AFT-3} for a class of potentials $q(x)$
satisfying different conditions
(without radial symmetry).
For a radially symmetric potential satisfying \eqref{hyp:growth:q(r)},
the anti-maximun principle has been obtained previously in
{Alziary}, {Fleckinger}, and {Tak\'a\v{c}}
\cite[Theorem 2.1, p.~128]{AFT-1} for $N=2$ and
\cite[Theorem 2.1, p.~365]{AFT-2} for $N\geq 2$.
Furthermore, in \cite{AFT-1, AFT-2}
the function $f$ is assumed to be
a ``sufficiently smooth'' perturbation of
a radially symmetric function from~$X$.

Theorem~\ref{thm-Antimax-q(x)} is an immediate consequence of
the spectral decomposition of the resolvent of $\mathcal{A}$ as
%
\begin{equation}
  (\lambda I - \mathcal{A})^{-1}
  = (\lambda - \Lambda)^{-1} \mathcal{P} + \mathcal{H}(\lambda)
    \quad\mbox{for }\quad
  0 < | \lambda - \Lambda | < \eta ,
\label{e:Laurent}
\end{equation}
%
see, e.g., {Sweers} \cite[Theorem 3.2(ii), p.~259]{Sweers}
or {Tak\'a\v{c}} \cite[Eq.~(6), p.~67]{Takac}.
Here, $\lambda\in \mathbb{C}$, $\eta > 0$ is small enough,
$\mathcal{P}$ denotes the spectral projection onto the eigenspace
spanned by $\varphi$, and
$\mathcal{H}(\lambda): L^2(\mathbb{R}^N)\to L^2(\mathbb{R}^N)$
is a holomorphic family of compact linear operators parametrized by
$\lambda$ with $| \lambda - \Lambda | < \eta$.
Moreover, $\mathcal{P}$ is selfadjoint and
%
\begin{math}
    \mathcal{P} \mathcal{H}(\lambda)
  = \mathcal{H}(\lambda) \mathcal{P}
  = 0
\end{math}
%
on $L^2(\mathbb{R}^N)$.
Formula \eqref{e:Laurent} is used to prove the anti\--maximum principle
also in
\cite[Eq.\ (6), p.~124]{AFT-1} and
\cite[Eq.\ (6), p.~361]{AFT-2}.
%
The main idea of the proof of Theorem~\ref{thm-Antimax-q(x)}
is to show that each of the linear operators
$\{ \mathcal{H}(\lambda): | \lambda - \lambda_1 | < \eta \}$
is bounded not only on $L^2(\mathbb{R}^N)$ but also on~$X$.
Clearly, given the Neumann series expansion of
$\mathcal{H}(\lambda)$ for $|\lambda - \Lambda| < \eta$,
it suffices to show that the restriction
$\mathcal{H}(\Lambda)\vert_X$ of
$\mathcal{H}(\Lambda)$ to $X$ is a bounded linear operator on $X$.
But this clearly follows from Theorem \ref{thm-X-compact}, Part~(c),
with a help from
formula~(6.32) in {Kato} \cite[Chapt.\ III, {\S}6.5, p.~180]{Kato}
or formula~(1) in {Yosida} \cite[Chapt.\ VIII, Sect.~8, p.~228]{Yosida}.

In various common versions of the anti\--maximum principle
in a bounded domain $\Omega\subset \mathbb{R}^N$, $N\geq 1$,
besides the assumption $0\leq f\not\equiv 0$ in $\Omega$,
it is only assumed that
$f\in L^p(\Omega)$ for some $p>N$
(cf.\ {Cl\'ement} and {Peletier} \cite[Theorem~1, p.~222]{ClemPel},
{Sweers} \cite{Sweers} or {Tak\'a\v{c}} \cite{Takac}).
For $\Omega = \mathbb{R}^N$
the authors \cite[Example 4.1, pp.\ 377--379]{AFT-2}
have constructed an example of a simple potential
$q(r)$ and a function $f(r)$, both radially symmetric,
$f\in L^2(\mathbb{R}^N)\setminus X$, and $0\leq f\not\equiv 0$ in $\mathbb{R}^N$,
in which even the inequality $u\leq 0$ a.e.\ in $\mathbb{R}^N$
(weaker than the anti\--maximum principle of
Theorem~\ref{thm-Antimax-q(x)})
is violated.
More precisely, if $|\lambda - \Lambda| > 0$ is small enough,
then even $u(r) > 0$ for every $r>0$ large enough.

\section{Preliminary results}
\label{s:Prelim}

In {Alziary}, {Fleckinger}, and {Tak\'a\v{c}} \cite{AFT-3},
an important ingredient in comparing the ground states
$\varphi_j(x)\equiv \varphi_{Q_j}(|x|)$
corresponding to the potentials $Q_j(|x|)$ ($j=1,2$)
is an asymptotic formula due to
{Hartman} and {Wintner} \cite[eq.\ (xxv), p.~49]{Hartman-Wint}.
In contrast, here we take advantage of
a generalized Titchmarsh' lemma (Lemma \ref{lem-Titchmarsh})
and a comparison result (Lemma \ref{lem-U1:U2})
to obtain the equivalence of the ground states,
$\varphi_1$ and $\varphi_2$, under Hypothesis \eqref{hyp:growth:q(x)}.

\subsection{Generalized Titchmarsh' lemma}
\label{ss:Titchmarsh}

Throughout this paragraph we assume that
$U: [R,\infty)\to (0,\infty)$
is monotone increasing and continuous, where
$0\leq R$ $< \infty$.
Hence, $U'(r)\geq 0$ for a.e.\ $r\geq R$.
We generalize Titchmarsh' lemma
(cf.\ {Titchmarsh} \cite[Sect.~8.2, p.~165]{Titchmarsh})
used in
{Alziary} and {Tak\'a\v{c}} \cite{AlziaryTak}, Lemma 3.2, p.~286.

\begin{lemma}\label{lem-Titchmarsh}
Assume that
$f, f': [R,\infty)\to \mathbb{R}$ are locally absolutely continuous,
$f(r) > 0$ for all  $r\geq R$, and  $f$ satisfies
%
\begin{equation}
  - f'' + U(r) f\leq 0 \quad\mbox{for a.e. } r\geq R .
\label{ineq:f,f'}
\end{equation}
%
If  $f'\leq 0$ on $[R,\infty)$ then we must have
%
\begin{equation}
 - {f'}/{f} \geq U(r)^{1/2} \quad\mbox{for all } r\geq R .
\label{ineq:g}
\end{equation}
%
\end{lemma}

\begin{proof}
Upon the substitution $g = - (\log f)' = - f' / f$, where
$g\geq 0$ on $[R,\infty)$,
inequality \eqref{ineq:f,f'} is equivalent to
%
\begin{equation}
  g'\leq g^2 - U(r) \quad\mbox{for all } r\geq R .
\label{ineq:g^2,g'}
\end{equation}
%
This follows from
%
\begin{math}
  g'= {}- {f''}/{f} + ({f'}/{f})^2 = {}- {f''}/{f} + g^2 .
\end{math}
%
Equivalently to \eqref{ineq:g}, we claim that
$g\geq U^{1/2}$ on $[R,\infty)$.

Indeed, if on the contrary
\[
  \delta\stackrel{{\rm def}}{=} g(r_0)^2 - U(r_0) < 0
  \quad\mbox{for some } r_0\geq R ,
\]
then from \eqref{ineq:g^2,g'}
and the continuity of $g$ and $U$ at $r_0$ we deduce that
there exists some $r_1 > r_0$ such that also
\[
  g'(r)\leq g(r)^2 - U(r) < 0 \quad\mbox{for a.e. } r\in [r_0, r_1) .
\]
Consequently, with regard to $U(r)\geq U(r_0)$ for $r\geq r_0$, we get
%
\begin{equation*}
  g'(r)\leq g(r)^2 - U(r) < g(r_0)^2 - U(r)\leq g(r_0)^2 - U(r_0)
  = \delta < 0
\end{equation*}
%
for a.e.\ $r\in (r_0, r_1]$.
Hence, we may take $r_1 > r_0$ arbitrarily large as long as
$g(r_1)\geq 0$ holds;
let us choose $r_1 > r_0$ so large that $g(r_1) = 0$.
Then
\[
  g'(r_1)\leq g(r_1)^2 - U(r_1) = - U(r_1) < 0
\]
which implies $g(r_1 + s) < 0$ for every $s>0$ small enough
and thus contradicts our hypothesis $g\geq 0$ on $[R,\infty)$.
Hence, we have proved
$g(r)^2 - U(r)\geq 0$ for all $r\geq R$, which entails
$-f'/f = g\geq U^{1/2}$ on $[R,\infty)$.
\end{proof}

\begin{corollary}\label{cor-Titchmarsh}
In the situation of  {\rm Lemma~\ref{lem-Titchmarsh}} above,
the function $f$ is decreasing, convex, and satisfies
$f(s)\searrow 0$ and  $f'(s)\nearrow 0$ as $s\nearrow \infty$.
Moreover, we have
%
\begin{equation}
    \frac{f(s)}{f(r)}
  \leq \exp\Big( - \int_r^s U(t)^{1/2} \,{\rm d}t \Big)
    \quad\mbox{whenever }\, R\leq r\leq s < \infty .
%
\label{ineq:f(s)/f(r)}
\end{equation}
%
\end{corollary}

\begin{proof}
First, we integrate inequality \eqref{ineq:g}
over the interval $[r,s]$ to get \eqref{ineq:f(s)/f(r)}.
Since $f(s) > 0$ for all $s\geq R$, and $U$ is monotone increasing,
inequality \eqref{ineq:f(s)/f(r)} forces
$f(s)\searrow 0$ as $s\nearrow \infty$.
Next, \eqref{ineq:f,f'} guarantees that $f$ is convex.
It follows that $f'(s)\nearrow 0$ as $s\nearrow \infty$.
\end{proof}

\subsection{Asymptotic equivalence of solutions}
\label{ss:phi_1/phi_2}

In this paragraph we compare positive solutions of
homogeneous Schr\"odinger equations (or inequalities)
in $[R,\infty)$ with different potentials (for $N=1$).
%
We start with a comparison result proved in
{M.~Hoffmann\--Ostenhof} et al.\
\cite[Lemma 3.2, p.~348]{HoffSwetina}.

\begin{lemma}\label{lem-U1:U2}
Let $U_1, U_2: [R,\infty)\to (0,\infty)$
be two continuous potentials satisfying
$0 < \mbox{\rm const} \leq U_1\leq U_2$ for  $r\geq R$,
where $0 < R < \infty$.
Let all  $f_1, f_2, f_1', f_2'\in L^2(R,\infty)$
be locally absolutely continuous in $[R,\infty)$,
and let also $f_1 > 0$ and  $f_2 > 0$ for  $r\geq R$.
Finally, assume that
\[
  -f_1'' + U_1(r) f_1\geq 0 \quad\mbox{and }\quad
  -f_2'' + U_2(r) f_2\leq 0
    \quad\mbox{for almost every }\ r > R .
\]
Then we have
\[
  \frac{f_1 }{f_2}\geq \frac{f_1(R)}{f_2(R)}
    \;\mbox{ and }\;
  \frac{f_1'}{f_1}\geq \frac{f_2'  }{f_2   }
    \quad\mbox{for every }\ r\geq R .
\]
%
\end{lemma}

In the next proposition we give
a sufficient upper bound for the perturbation
$Q_2(r) - Q_1(r)$ in  \eqref{hyp:growth:q(x)}
that guarantees that all ground states
$\varphi_{q}$, $\varphi_{Q_1}$, and $\varphi_{Q_2}$ are comparable
(see {\S}\ref{ss:Compact-q(x)}).

\begin{proposition}\label{prop-Titchmarsh}
Let $U_1, U_2: [R,\infty)\to (0,\infty)$
be monotone increasing and continuous, where
$0\leq R < \infty$.
Assume that
$f_j, f_j': [R,\infty)\to \mathbb{R}$
are locally absolutely continuous; $j=1,2$,
$f_j(r) > 0$ and  $f_j'(r)\leq 0$ for all  $r\geq R$, and
$f_j$ satisfies
%
\begin{equation}
  - f_j'' + U_j(r) f_j = 0 \quad\mbox{for a.e. } r\geq R .
\label{eq:f_j,f_j'}
\end{equation}
%
Then for all  $r\geq R$ we have
%
\begin{equation} \label{ineq:log(f_1/f_2)}
\begin{split}
&  \left\vert
  \frac{{\rm d}}{\mathrm{d}r} \left( \log\, \frac{f_1(r)}{f_2(r)} \right)
    \right\vert \\
&\leq \int_r^\infty | U_1(s) - U_2(s) |\, \exp
    \Big( - \int_r^s [ U_1(t)^{1/2} + U_2(t)^{1/2} ] \,\mathrm{d}t
    \Big) \,\mathrm{d}s \,.
\end{split}
\end{equation}
%
\end{proposition}

\begin{proof}
Employing \eqref{eq:f_j,f_j'} we compute
\[
  \frac{{\rm d}}{{\rm d}r}
  \left[ f_1 f_2\left( \frac{f_1'}{f_1} - \frac{f_2'}{f_2} \right)
  \right]
  = f_1'' f_2 - f_1 f_2''
  = ( U_1(r) - U_2(r) )\, f_1 f_2 .
\]
Since
$f_1'(r) f_2(r)\to 0$ and $f_1(r) f_2'(r)\to 0$ as $r\to \infty$,
by Corollary~\ref{cor-Titchmarsh},
integration yields
\[
  - f_1(r) f_2(r)\left( \frac{f_1'}{f_1} - \frac{f_2'}{f_2} \right)
  = \int_r^\infty ( U_1(s) - U_2(s) )\, f_1(s) f_2(s) \,{\rm d}s ,
\]
i.e.,
%
\begin{equation*}
  \frac{\mathrm{d}}{\mathrm{d}r} \left( \log\, \frac{f_1}{f_2} \right)
  = {}- \frac{1}{ f_1(r) f_2(r) }
    \int_r^\infty ( U_1(s) - U_2(s) )\, f_1(s) f_2(s) \,\mathrm{d}s
\end{equation*}
%
for all $r\geq R$.
Consequently,
%
\begin{equation*}
    \left\vert \frac{\mathrm{d}}{\mathrm{d}r}
      \left( \log\, \frac{f_1(r)}{f_2(r)} \right)
    \right\vert
  \leq \frac{1}{ f_1(r) f_2(r) }
    \int_r^\infty | U_1(s) - U_2(s) |\, f_1(s) f_2(s) \,\mathrm{d}s
\end{equation*}
%
for all $r\geq R$.
Finally, we apply inequality \eqref{ineq:f(s)/f(r)}
with $U = U_j$ and $f = f_j$; $j=1,2$, to the right\--hand side of
the inequality above to get \eqref{ineq:log(f_1/f_2)}.
\end{proof}


\begin{remark}\label{rem-Titchmarsh}\rm
In the situation of  {\rm Proposition~\ref{prop-Titchmarsh}} above,
the functions $f_1(r)$ and  $f_2(r)$
are \emph{asymptotically equivalent near infinity },
meaning that the ratios
${f_1(r)}/{f_2(r)}$ and ${f_2(r)}/{f_1(r)}$
stay bounded as $r\to \infty$,
provided the following condition is satisfied:
%
\begin{equation}
\label{e:int-diff:U_2-U_1}
\begin{split}
&   \int_R^\infty \int_r^\infty | U_1(s) - U_2(s) |\,
  \exp\Big( - \int_r^s [ U_1(t)^{1/2} + U_2(t)^{1/2} ] \,{\rm d}t \Big)
  \,{\rm d}s \,{\rm d}r =
\\
&   \int_R^\infty | U_1(s) - U_2(s) | \int_R^s
  \exp\Big( - \int_r^s [ U_1(t)^{1/2} + U_2(t)^{1/2} ] \,{\rm d}t \Big)
  \,{\rm d}r \,{\rm d}s
  < \infty \,.
\end{split}
\end{equation}
%
This claim follows by integrating \eqref{ineq:log(f_1/f_2)}
with respect to $r\in [R,\infty)$.
Note that condition \eqref{e:int-diff:U_2-U_1} corresponds to
\eqref{e:int-diff} for $Q_1$ and $Q_2$.
%
\end{remark}


Simple sufficient conditions for $U_1$ and $U_2$ are given in
the Appendix, {\S}\ref{ss:power-grow}, which guarantee that
\eqref{e:int-diff:U_2-U_1} is satisfied.


\section{Two known compactness results}
\label{s:AFT-compact}

We state in this section two compactness results obtained in
{Alziary}, {Fleckin\-ger}, and {Tak\'a\v{c}}
\cite[Section~6]{AFT-3}
which are essential for the proof of our main Theorem \ref{thm-X-compact}.
We need local compactness from Proposition \ref{prop-X^o-B_R}
together with a compactness result by comparison of two potentials from
Proposition \ref{prop-q_1<q_2}.
These propositions are valid for any potential satisying \eqref{cond:q_0}
and, thus, we refer to \cite[Sections 6 and~8]{AFT-3} for the proofs.

\subsection{A local compactness result}
\label{ss:loc-compact}

The potential $q$ is assumed to satisfy only
conditions \eqref{cond:q_0} in this section.
If $B_R(0)$ is an open ball of radius $R$ ($0 < R < \infty$)
in $\mathbb{R}^N$ centered at the origin,
let $u\vert_{B_R(0)}$ denote the restriction of a function
$u: \mathbb{R}^N\to \mathbb{R}$ to $B_R(0)$.

\begin{proposition}\label{prop-X^o-B_R}
Assume that  $q: \mathbb{R}^N\to \mathbb{R}$ is a continuous function
satisfying \eqref{cond:q_0}, and let  $\lambda < \Lambda$.
Then, given any  $0 < R < \infty$, the restricted resolvent
%
\begin{equation*}
  \mathcal{R}_R: X^\odot\to L^1(B_R(0))
               : f\mapsto u\vert_{B_R(0)}
\end{equation*}
%
is compact, where
$u = (\mathcal{A} - \lambda I)^{-1}\vert_{X^\odot} f$.
%
\end{proposition}

This proposition is proved in \cite[Theorem 6.1]{AFT-3}.

\subsection{Compactness by comparison of two potentials}
\label{ss:compact-q_1<q_2}

Let us consider two potentials,
$q_j: \mathbb{R}^N\to \mathbb{R}$ for $j=1,2$,
each assumed to be a continuous function satisfying
only conditions \eqref{cond:q_0} in place of $q$.
We denote by $\Lambda_j = \Lambda_{q_j}$
the principal eigenvalue of the Schr\"odinger operator
%
\begin{equation}
  \mathcal{A}_j = \mathcal{A}_{q_j}\stackrel{{\rm def}}{=} - \Delta + q_j(x)\bullet
  \quad\mbox{on }\, L^2(\mathbb{R}^N) .
\label{op_j:Schroed_lam}
\end{equation}
%
The associated eigenfunction
$\varphi_j = \varphi_{q_j}$ is normalized by
$\varphi_j > 0$ throughout $\mathbb{R}^N$ and
$\|\varphi_j\|_{L^2(\mathbb{R}^N)} = 1$.
Finally, we write
$X_j = X_{q_j}$ and
$X_j^\odot = L^1(\mathbb{R}^N; \varphi_j \,{\rm d}x)$.

The following comparison result is natural
(and holds without any growth conditions other than \eqref{cond:q_0}).

\begin{proposition}\label{prop-q_1<q_2}
Assume $q_1\leq q_2$ in $\mathbb{R}^N$.
Then the following statements hold:
%
\begin{enumerate}
%
\item[{\rm (a)}]
%
$0 < \Lambda_1\leq \Lambda_2 < \infty$.
%
\item[{\rm (b)}]
%
For each $\lambda < \Lambda_1$,
%
\begin{equation*}
  f\geq 0 \;\mbox{ in } L^2(\mathbb{R}^N)
    \quad\Longrightarrow\quad
  (\mathcal{A}_{q_2} - \lambda I)^{-1} f\leq
  (\mathcal{A}_{q_1} - \lambda I)^{-1} f
    \;\mbox{ in } L^2(\mathbb{R}^N) .
\end{equation*}
%
\item[{\rm (c)}]
%
Given any  $\lambda < \Lambda_1$, if the restriction
%
\begin{math}
  (\mathcal{A}_{q_1} - \lambda I)^{-1}\vert_{X_1}
  : X_1\to X_1
\end{math}
%
of the resolvent
$(\mathcal{A}_{q_1} - \lambda I)^{-1}$
to $X_1$ is weakly compact, then
$(\mathcal{A}_{q_j} - \lambda I)^{-1}\vert_{X_1}$
is also compact for  $j=1,2$.
%
\item[{\rm (c')}]
%
Given any  $\lambda < \Lambda_1$, if the extension
%
\begin{math}
  (\mathcal{A}_{q_1} - \lambda I)^{-1}\vert_{X_1^\odot}
  : X_1^\odot\to X_1^\odot
\end{math}
%
of the resolvent
$(\mathcal{A}_{q_1} - \lambda I)^{-1}$
to $X_1^\odot$ is weakly compact, then
$(\mathcal{A}_{q_j} - \lambda I)^{-1}\vert_{X_1^\odot}$
is also compact for  $j=1,2$.
%
\end{enumerate}
%
\end{proposition}

\begin{corollary}\label{cor-q_1<q_2}
Assume $q_1\leq q_2$ in $\mathbb{R}^N$.
If the weak compactness condition (the ``if'' part)
in {\rm (c)} or  {\rm (c')}, {\rm Proposition~\ref{prop-q_1<q_2}},
is satisfied, for some $\lambda < \Lambda_1$, then we have
$\sup_{\mathbb{R}^N} (\varphi_2 / \varphi_1) < \infty$ or, equivalently,
$X_2\hookrightarrow X_1$ is a continuous embedding.
%
\end{corollary}


 Proposition \ref{prop-q_1<q_2} and
 Corollary \ref{cor-q_1<q_2} are proved in
{Alziary}, {Fleckinger}, and {Tak\'a\v{c}} \cite{AFT-3},
Proposition 8.1 and Corollary 8.2, respectively.
The proof of Proposition \ref{prop-q_1<q_2} makes use of
Proposition \ref{prop-X^o-B_R}.

\section{Compactness for potentials of class \eqref{class:Q(r)}}
\label{s:Compact-Q(r)}

Throughout this section we consider only a radially symmetric potential
$q$ of class \eqref{class:Q(r)},
$q(x) = Q(|x|)$ for all $x\in \mathbb{R}^N$.
All symbols
$\mathcal{A}$, $\Lambda$, $\varphi$, $X$, $X^\odot$, etc.
are considered only for this special type of potential.
Under the hypotheses in \eqref{class:Q(r)},
we are able to show the following special case of
Theorem \ref{thm-X-compact}, Part~(a):

\begin{proposition}\label{prop-X-compact}
Both operators $(\mathcal{A} - \lambda I)^{-1}\vert_{X} : X\to X$
and
%
\begin{math}
  (\mathcal{A} - \lambda I)^{-1}\vert_{X^\odot}
  : X^\odot\to X^\odot
\end{math}
%
are compact.
%
\end{proposition}

Also the compactness of
$(\mathcal{A} - \lambda I)^{-1}\vert_{X} : X\to X$
is equivalent to that of
%
\begin{math}
  (\mathcal{A} - \lambda I)^{-1}\vert_{X^\odot}
  : X^\odot\to X^\odot ,
\end{math}
%
by Schauder's theorem
({Edwards} \cite[Corollary 9.2.3, p.~621]{Edwards} or
 {Yosida} \cite[Chapt.~X, Sect.~4, p.~282]{Yosida});
we will prove the latter one.

We split the proof of Proposition \ref{prop-X-compact}
into two paragraphs,
{\S}\ref{ss:Compact-rad} and {\S}\ref{ss:Compact-gen}.
We set
$\mathcal{K} = (\mathcal{A} - \lambda I)^{-1}$ on $L^2(\mathbb{R}^N)$.
In the first paragraph, {\S}\ref{ss:Compact-rad},
we restrict the operators
$\mathcal{K}\vert_{X}$ and $\mathcal{K}\vert_{X^\odot}$
to the corresponding subspaces of radially symmetric functions
and show that Proposition \ref{prop-X-compact}
is valid in these subspaces; see
Lemmas \ref{lem-X-compact} and \ref{lem-X^o-compact} below.
In the second paragraph, {\S}\ref{ss:Compact-gen},
we take advantage of Lemma \ref{lem-X^o-compact}
to prove the compactness of $\mathcal{K}\vert_{X^\odot}$
in Proposition \ref{prop-X-compact}.

\subsection{Compactness on the space of radial functions}
\label{ss:Compact-rad}

Throughout this paragraph, we denote by
$X_\mathrm{rad}$, $L_\mathrm{rad}^2(\mathbb{R}^N)$, and $X_\mathrm{rad}^\odot$,
respectively, the subspaces of
$X$, $L^2(\mathbb{R}^N)$, and $X^\odot$ that consist of
all radially symmetric functions from these spaces.
All these subspaces are closed.
Moreover, since the potential $Q$ is radially symmetric,
all subspaces above are invariant under the operator
$\mathcal{K}\vert_{X^\odot}$.
We denote by
$\mathcal{K}\vert_{X_\mathrm{rad}}$,
$\mathcal{K}\vert_{ L_\mathrm{rad}^2(\mathbb{R}^N) }$, and
$\mathcal{K}\vert_{ X_\mathrm{rad}^\odot }$, respectively,
the restrictions of
$\mathcal{K}\vert_{X^\odot}$ to the spaces
$X_\mathrm{rad}$, $L_\mathrm{rad}^2(\mathbb{R}^N)$, and $X_\mathrm{rad}^\odot$.
These restrictions have similar properties as
$\mathcal{K}\vert_{X}$, $\mathcal{K}$, and
$\mathcal{K}\vert_{X^\odot}$, respectively, above.

\begin{lemma}\label{lem-X-compact}
Under the hypotheses in \eqref{class:Q(r)}
the operator
%
\begin{math}
  \mathcal{K}\vert_{X_\mathrm{rad}}:
  X_\mathrm{rad}\to X_\mathrm{rad}
\end{math}
%
is compact.
%
\end{lemma}

By Schauder's theorem again
({Edwards} \cite[Corollary 9.2.3, p.~621]{Edwards} or
 {Yosida} \cite[Chapt.~X, Sect.~4, p.~282]{Yosida}),
this lemma is equivalent to

\begin{lemma}\label{lem-X^o-compact}
Under the hypotheses in \eqref{class:Q(r)}
the operator
%
\begin{math}
  \mathcal{K}\vert_{ X_\mathrm{rad}^\odot }:
  X_\mathrm{rad}^\odot \to X_\mathrm{rad}^\odot
\end{math}
%
is compact.
%
\end{lemma}

The following simple consequence of
 \eqref{hyp:growth:q(r)}
on the growth of the ground state $\varphi$
is needed in the proof of Lemma \ref{lem-X-compact}:

\begin{lemma}\label{lem-growth:phi(r)}
Under  \eqref{hyp:growth:q(r)} the function
$P: (r_0,\infty)\to (0,\infty)$,
defined by $P(r) = 2\, Q(r)^{1/2}$ for  $r > r_0$,
is monotone increasing,
$\int_{r_0}^\infty P(r)^{-1} \,{\rm d}r$ $< \infty$, and
%
\begin{equation}
    p(r)\stackrel{{\rm def}}{=}
{}- \frac{N-1}{r} - 2 (\log\varphi(r))' \geq P(r) > 0
    \quad\mbox{holds for all } r > r_0 .
\label{growth:phi(r)}
\end{equation}
%
\end{lemma}

\begin{proof}
Using the radial Schr\"odinger equation for $\varphi$,
%
\begin{equation*}
  - \varphi''(r) - \frac{N-1}{r}\, \varphi'(r) + q(r) \varphi(r)
  = \Lambda \varphi(r)
  \quad\mbox{for } r\geq R ,
\end{equation*}
%
we observe that the function
$v(r) = r^{(N-1)/2} \varphi(r) > 0$ must satisfy
%
\begin{equation*}
  - v''(r)
  + \left( q(r) - \Lambda + \frac{(N-1)(N-3)}{4 r^2} \right) v(r)
  = 0
  \quad\mbox{for } r\geq R .
\end{equation*}
%
It follows from \eqref{growth:q(r)} that
%
\begin{equation*}
  - v''(r) + Q(r) v(r)\leq 0 \quad\mbox{for } r\geq R .
\end{equation*}
%

Next, we claim that $v'(r)\leq 0$ for all $r\geq R$.
Indeed, since $v''(r)\geq Q(r) v(r)\geq 0$,
the derivative $v'$ is nondecreasing on $[R,\infty)$.
Therefore, if $v'(r_0) > 0$ for some $r_0\geq R$,
then $v'(r)\geq v'(r_0) > 0$ for every $r\geq r_0$,
which contradicts
$\int_{r_0}^\infty v(r)\, r^{(N-1)/2} \,{\rm d}r < \infty$.
Finally, we may apply
the generalized Titchmarsh' lemma to conclude that the function
$g = - (\log v)' = - v' / v$ satisfies
$g(r)\geq Q(r)^{1/2}$ for all $r\geq R$.
We compute
\[
  g(r) = {}- \frac{{\rm d}}{{\rm d}r}
    \log \left( r^{(N-1)/2} \varphi(r) \right)
  = {}- \frac{N-1}{2r} - \frac{{\rm d}}{{\rm d}r}\, \log \varphi(r)
  = \textstyle\frac12 \, p(r)
\]
with $g(r)\geq Q(r)^{1/2}$ for all $r\geq R$.
This proves \eqref{growth:phi(r)}.
The remaining claims follow from the properties of
class \eqref{class:Q(r)}.
\end{proof}


We prove Lemma \ref{lem-X-compact} directly using
Arzel\`a\--Ascoli's compactness criterion for
continuous functions on the one point compactification
$\mathbb{R}_+^* = \mathbb{R}_+\cup \{\infty\}$ of $\mathbb{R}_+$.
The metric on $\mathbb{R}_+^*$ is defined by
%
\begin{equation*}
  d(x,y)\stackrel{{\rm def}}{=} \begin{cases}
  \frac{|x-y|}{1 + |x-y|} & \mbox{for } x,y\in \mathbb{R}_+ ; \\
  1 & \mbox{for } 0\leq x < y = \infty \;\mbox{ or }\;
                           0\leq y < x = \infty ;\\
  0 & \mbox {for }\; x = y = \infty .
\end{cases}
\end{equation*}
%
We denote by $C(\mathbb{R}_+^*)$
the Banach space of all continuous functions on
the compact metric space $\mathbb{R}_+^*$ endowed with
the supremum norm from $L^\infty(\mathbb{R}_+)$.

\begin{proof}[Proof of Lemma~\ref{lem-X-compact}]
Given $f,u\in X_\mathrm{rad}$, $u = \mathcal{K} f$
is equivalent with the ordinary differential equation
%
\begin{equation*}
  - u''(r) - \frac{N-1}{r}\, u'(r) + q(r) u(r)
  = \lambda u(r) + f(r)
  \quad\mbox{for } 0 < r < \infty
%\label{eq:u=Kf}
\end{equation*}
%
supplemented by the conditions
%
\begin{equation*}
  \lim_{r\to 0+} u'(r) = 0 \quad\mbox{and }\quad
       \sup_{0 < r < \infty} \genfrac{|}{|}{}0{u(r)}{\varphi(r)}
  \leq (\Lambda - \lambda)^{-1} \cdot
       \sup_{0 < r < \infty} \genfrac{|}{|}{}0{f(r)}{\varphi(r)} \,.
%\label{bc:u=Kf}
\end{equation*}
%
Clearly,
the former one is a boundary condition at zero
that follows from the radial symmetry,
whereas the latter one follows from the weak maximum principle.

Substituting
$g = f / \varphi$ and $v = u / \varphi$, combined with
%
\begin{equation*}
  - \varphi''(r) - \frac{N-1}{r}\, \varphi'(r) + q(r) \varphi(r)
  = \Lambda \varphi(r)
  \quad\mbox{for } 0 < r < \infty ,
\end{equation*}
%
we have equivalently
%
\begin{equation}
  - v''(r) - \frac{N-1}{r}\, v'(r) - 2 (\log\varphi(r))'\, v'(r)
  + (\Lambda - \lambda) v(r) = g(r)
  \quad\mbox{for } 0 < r < \infty
\label{eq:v}
\end{equation}
%
subject to the conditions
%
\begin{equation}
  \lim_{r\to 0+} v'(r) = 0 \quad\mbox{and }\quad
       \sup_{0 < r < \infty} |v(r)|
  \leq (\Lambda - \lambda)^{-1} \cdot
       \sup_{0 < r < \infty} |g(r)| \,.
\label{cond:v}
\end{equation}
%
Then $\mathcal{K}\vert_{X_\mathrm{rad}}$ is compact on $X_\mathrm{rad}$
if and only if the linear operator
$\mathcal{K}_{\varphi} : L^\infty(\mathbb{R}_+)\to L^\infty(\mathbb{R}_+)$,
defined by
\[
    \mathcal{K}_{\varphi} g\stackrel{{\rm def}}{=} v
  = \varphi^{-1}\cdot \mathcal{K} (g\varphi)
  \quad\mbox{for } g\in L^\infty(\mathbb{R}_+) ,
\]
is compact.

We will apply Arzel\`a\--Ascoli's compactness criterion
in the Banach space $C(\mathbb{R}_+^*)$ in order to show that the image
%
\begin{math}
  \mathcal{K}_{\varphi}
  \left( \overline{\mathcal{B}}_{ L^\infty(\mathbb{R}_+) } \right)
\end{math}
%
of the unit ball
\[
  \overline{\mathcal{B}}_{ L^\infty(\mathbb{R}_+) }
  = \left\{ g\in L^\infty(\mathbb{R}_+):
            \| g\|_{ L^\infty(\mathbb{R}_+) } \leq 1
    \right\}
\]
has compact closure in $C(\mathbb{R}_+^*)$.
Since $L^\infty(\mathbb{R}_+)$ is a Banach lattice, it suffices to show that
%
\begin{math}
  \mathcal{K}_{\varphi}
  \Big( \overline{\mathcal{B}}_{ L^\infty(\mathbb{R}_+) }^+ \Big)
\end{math}
%
has compact closure in $C(\mathbb{R}_+^*)$, where
\[
  \overline{\mathcal{B}}_{ L^\infty(\mathbb{R}_+) }^+
  = \left\{ g\in \overline{\mathcal{B}}_{ L^\infty(\mathbb{R}_+) } :
            g\geq 0 \,\mbox{ in }\, \mathbb{R}_+
    \right\} .
\]
Clearly, the function $v$ from
\eqref{eq:v} and \eqref{cond:v} above satisfies $v\in C^1(\mathbb{R}_+)$;
we will show also $v\in C(\mathbb{R}_+^*)$.
Therefore, we need to show that the linear operator
%
\begin{equation*}
  \mathcal{K}_{\varphi} :
  L^\infty(\mathbb{R}_+)\to C(\mathbb{R}_+^*)\subset L^\infty(\mathbb{R}_+)
\end{equation*}
%
is compact.

So let $g\in L^\infty(\mathbb{R}_+)$ be arbitrary with
$0\leq g(r)\leq 1$ for $r\in \mathbb{R}_+$.
Hence,
$v = \mathcal{K}_{\varphi} g$ satisfies
$v\in C^1(\mathbb{R}_+)$ and also
$0\leq v(r)\leq (\Lambda - \lambda)^{-1}$,
by \eqref{e:max-princ_X}.
It follows that the function
%
\begin{equation*}
  g^\sharp\stackrel{{\rm def}}{=} g - (\Lambda - \lambda) v
\end{equation*}
%
satisfies $-1\leq g^\sharp\leq 1$, and the derivative
$w\stackrel{{\rm def}}{=} v'$ verifies the ordinary differential equation
%
\begin{equation}
  - w'(r) - \frac{N-1}{r}\, w(r) - 2 (\log\varphi(r))'\, w(r)
  = g^\sharp(r)
  \quad\mbox{for } 0 < r < \infty
\label{eq:w}
\end{equation}
%
subject to the conditions
%
\[
 \lim_{r\to 0+} w(r) = 0 \quad\mbox{and }\quad
       \sup_{0 < r < \infty} \left| \int_0^r w(s) \,{\rm d}s \right|
  \leq (\Lambda - \lambda)^{-1} \,.
%\label{cond:w}
\]
%
The latter condition has been obtained from
%
\[
 \int_0^r w(s) \,{\rm d}s = v(r) - v(0)
    \quad\mbox{with }\quad
  0\leq v(r)\leq (\Lambda - \lambda)^{-1}
\]
%
for all $r\geq 0$.
Since $w$ is continuous, this condition implies that
there exists a sequence
$\{ r_n\}_{n=1}^\infty \subset \mathbb{R}_+$ such that
$r_n\to \infty$ and $w(r_n)\to 0$ as $n\to \infty$.

The differential equation \eqref{eq:w} is equivalent to
%
\begin{equation*}
- \frac{{\rm d}}{{\rm d}r}
    \left( r^{N-1} \varphi(r)^2\, w(r) \right)
  = r^{N-1} \varphi(r)^2\, g^\sharp(r)
  \quad\mbox{for } 0 < r < \infty .
%\label{eq:w:equiv}
\end{equation*}
%
After integration, we thus arrive at
%
\[
  r^{N-1} \varphi(r)^2\, w(r)
  - s^{N-1} \varphi(s)^2\, w(s)
  = \int_{r}^{s} t^{N-1} \varphi(t)^2\, g^\sharp(t) \,{\rm d}t
%\label{eq:w:int_r^s}
\]
%
whenever $0\leq r,s < \infty$.
Applying $\lim_{s\to 0+} w(s) = 0$ we obtain
%
\begin{equation}
   r^{N-1} \varphi(r)^2\, w(r)
  = - \int_{0}^{r} t^{N-1} \varphi(t)^2\, g^\sharp(t) \,{\rm d}t
  \quad\mbox{for all } r\geq 0 .
\label{eq:w:int_0^r}
\end{equation}
%
Taking $s = r_n$ and letting $n\to \infty$ we obtain also
%
\begin{equation}
   r^{N-1} \varphi(r)^2\, w(r)
  = \int_{r}^{\infty} t^{N-1} \varphi(t)^2\, g^\sharp(t) \,{\rm d}t
  \quad\mbox{for all } r\geq 0 .
\label{eq:w:int_r}
\end{equation}
%
Here we have used the facts that
$s^{N-1} \varphi(s)^2 \to 0$ as $s\to \infty$ together with
$r_n\to \infty$ and $w(r_n)\to 0$ as $n\to \infty$.
Recall the normalization
%
\begin{math}
  \int_0^\infty \varphi(t)^2\, t^{N-1} \,{\rm d}t
  = \sigma_{N-1}^{-1} ,
\end{math}
%
where $\sigma_{N-1}$ stands for
the surface area of the unit sphere in $\mathbb{R}^N$.
Below we will take advantage of formulas
\eqref{eq:w:int_0^r} and \eqref{eq:w:int_r}
to estimate $|w(r)|$ as $r\to 0+$ and $r\to \infty$, respectively.

Because of $|g^\sharp|\leq 1$, eq.~\eqref{eq:w:int_0^r} yields
$|w|\leq w^\sharp_0$ where
$w^\sharp_0: \mathbb{R}_+\to \mathbb{R}_+$ is the function defined by
$w^\sharp_0(0) = 0$ and
%
\[
  r^{N-1} \varphi(r)^2\, w^\sharp_0(r)
  = \int_{0}^{r} t^{N-1} \varphi(t)^2 \,{\rm d}t
  \quad\mbox{for all } r > 0 .
%\label{eq:^w_n:int_0^r}
\]
%
Using
$\lim_{r\to 0+} \varphi(r) = \varphi(0) > 0$
we conclude that
%
\begin{equation*}
%\label{est:^w_0:int_0^r}
\begin{split}
  \lim_{r\to 0+} \,\frac{w^\sharp_0(r)}{r}
& = \lim_{r\to 0+} \frac{1}{r} \int_{0}^{r}
    \genfrac{(}{)}{}0{t}{r}^{N-1}
    \genfrac{(}{)}{}0{\varphi(t)}{\varphi(r)}^2 \,{\rm d}t
\\
& = \lim_{r\to 0+} \frac{1}{r} \int_{0}^{r}
    \genfrac{(}{)}{}0{t}{r}^{N-1} \,{\rm d}t
  = \frac{1}{N}\, .
\end{split}
\end{equation*}
%

Because of $|g^\sharp|\leq 1$, eq.~\eqref{eq:w:int_r} yields
$|w|\leq w^\sharp_\infty$ where
$w^\sharp_\infty: (0,\infty)\to \mathbb{R}_+$
is the function defined by
%
\begin{equation}
   w^\sharp_\infty(r)
  = r^{-(N-1)} \varphi(r)^{-2}
    \int_{r}^{\infty} t^{N-1} \varphi(t)^2 \,{\rm d}t
  \quad\mbox{for all } r > 0 .
\label{eq:^w_infty:int_r}
\end{equation}
%
Next, we wish to show that
$w^\sharp_\infty(r)\leq P(r)^{-1}$ holds for all $r > r_0$, where
$P(r) = 2\, Q(r)^{1/2}$.
To this end, notice first that eq.~\eqref{eq:^w_infty:int_r}
is equivalent with
%
\begin{equation*}
    w^\sharp_{\infty}(r)
  = \int_{r}^{\infty}
    \exp\Big( - \int_r^s p(t) \,{\rm d}t \Big)
    \,{\rm d}s , \quad r > r_0 ,
\end{equation*}
%
where $p(t)$ is given by formula \eqref{growth:phi(r)}.
In particular, the function $w^\sharp_\infty(r)$
satisfies the differential equation
%
\begin{equation*}
- \frac{{\rm d}}{{\rm d}r}\, w^\sharp_{\infty}(r)
  + p(r)\, w^\sharp_\infty(r)
  = 1
  \quad\mbox{for } r_0 < r < \infty .
\end{equation*}
%
By Lemma \ref{lem-growth:phi(r)}, we have
$p(t)\geq P(t)\geq P(r)$ whenever $r_0 < r\leq t < \infty$.
Hence, we can estimate
%
\begin{equation*}
\begin{split}
    w^\sharp_{\infty}(r)
& \leq
    \int_r^{\infty}
    \exp\Big( - \int_r^s P(r) \,{\rm d}t \Big)
    \,{\rm d}s
\\
& = \int_r^{\infty} e^{ - P(r)\, (s-r) } \,{\rm d}s
  = \int_0^{\infty} e^{ - P(r)\, s } \,{\rm d}s
  = P(r)^{-1} .
\end{split}
\end{equation*}
%

To summarize our estimates for the functions
$w^\sharp_0: \mathbb{R}_+\to \mathbb{R}_+$ and
$w^\sharp_\infty: (0,\infty)\to \mathbb{R}_+$
in the inequalities
$|w|\leq w^\sharp_0$ for $r\geq 0$ and
$|w|\leq w^\sharp_\infty$ for $r > r_0$,
we observe that both functions
$w^\sharp_0$ and $w^\sharp_\infty$ are continuously differentiable
and satisfy the estimates
%
\begin{gather}
 |w(r)|\leq w^\sharp_0(r)\leq C\, r
  \quad\mbox{for }\; 0\leq r\leq r_0 ,
\label{est:^w_0:r<r_0}
\\
 |w(r)|\leq w^\sharp_\infty(r)\leq P(r)^{-1}
  \quad\mbox{for }\; r_0 < r < \infty ,
\label{est:^w_infty:r>r_0}
\end{gather}
%
where $C>0$ is a constant.
Recall that
$\int_{r_0}^\infty P(r)^{-1} \,{\rm d}r < \infty$,
by condition \eqref{e:int-Q^-1/2}.

Consequently, for $g$ ranging over
$L^\infty(\mathbb{R}_+)$ with $0\leq g\leq 1$ in $\mathbb{R}_+$,
the set of functions
$v = \mathcal{K}_{\varphi} g\in C^1(\mathbb{R}_+)$
defined above is {\it uniformly equicontinuous } on
the compact metric space $\mathbb{R}_+^*$, thanks to
%
\[
 v(r) = v(0) + \int_0^r w(s) \,{\rm d}s
       = v(\infty) - \int_r^\infty w(s) \,{\rm d}s
  \quad\mbox{for } 0\leq r < \infty .
\]
%
The limit
$v(\infty) = \lim_{r\to \infty} v(r) \in \mathbb{R}$
exists by \eqref{est:^w_infty:r>r_0}.
Furthermore, owing to
$0\leq v(r)\leq (\Lambda - \lambda)^{-1}$ for $r\in \mathbb{R}_+$,
this set is also {\it uniformly bounded } on $\mathbb{R}_+^*$.
Thus, by Arzel\`a\--Ascoli's compactness criterion, the set
%
\begin{math}
  \mathcal{K}_{\varphi}
  \Big( \overline{\mathcal{B}}_{ L^\infty(\mathbb{R}_+) }^+ \Big)
\end{math}
%
has compact closure in $C(\mathbb{R}_+^*)$.

We have proved that the linear operator
$\mathcal{K}\vert_{X_\mathrm{rad}}$ is compact on $X_\mathrm{rad}$
and, moreover, its image satisfies
$\mathcal{K}(X_\mathrm{rad}) \subset C(\mathbb{R}_+^*)$.
\end{proof}

\subsection{Compactness on the entire space $X$}
\label{ss:Compact-gen}

We keep the assumption
$q(x) = Q(|x|)$ for all $x\in \mathbb{R}^N$.
Recall $\lambda < \Lambda$ and
$\mathcal{K} = (\mathcal{A} - \lambda I)^{-1}$ on $L^2(\mathbb{R}^N)$.
This time we will show first that the operator
$\mathcal{K}\vert_{X^\odot}$ is compact on $X^\odot$.
We derive this result from the compactness of its restriction
$\mathcal{K}\vert_{ X_\mathrm{rad}^\odot }$ to $X_\mathrm{rad}^\odot$
which we have already established in the previous paragraph.

To prove the compactness of $\mathcal{K}\vert_{X^\odot}$,
we will apply the well\--known compactness criterion of
Fr\'echet and Kolmogorov in the Lebesgue space
$X^\odot =$\break $L^1(\mathbb{R}^N; \varphi \,{\rm d}x)$;
see {Edwards} \cite[Theorem 4.20.1, p.~269]{Edwards}
or  {Yosida} \cite[Chapt.~X, Sect.~1, p.~275]{Yosida}.
We denote by
\[
  \overline{\mathcal{B}}_{X^\odot}
  = \left\{ f\in X^\odot: \| f\|_{X^\odot} \leq 1
    \right\}
\]
the closed unit ball centered at the origin in the Banach lattice
$X^\odot = L^1(\mathbb{R}^N; \varphi \,\mathrm{d}x)$.


\begin{lemma}\label{lem-X^o-infty}
Given any $\varepsilon > 0$, there exists a number
$R\equiv R(\varepsilon)\in (0,\infty)$ such that for every
$f\in \overline{\mathcal{B}}_{X^\odot}$ and
$u = \mathcal{K}\vert_{X^\odot} f$ we have
%
\begin{equation}
 \int_{|x|\geq R} |u(x)|\, \varphi(|x|) \,{\rm d}x \leq \varepsilon .
\label{est:int_|u|<eps}
\end{equation}
%
\end{lemma}

The proof of this lemma is given in
{Alziary}, {Fleckinger}, and {Tak\'a\v{c}}
\cite{AFT-3}, proof of Lemma 7.4, where \cite[Lemma 7.3]{AFT-3}
has to be replaced by our Lemma~\ref{lem-X^o-compact}.

\begin{proof}[Proof of Proposition~\ref{prop-X-compact}]
It suffices to prove that $\mathcal{K}\vert_{X^\odot}$ is compact.
Let $0 < R < \infty$.
Since the restricted resolvent
%
\begin{math}
  \mathcal{R}_R: X^\odot\to L^1(B_R(0))
               : f\mapsto u\vert_{B_R(0)}
\end{math}
%
is compact, by Proposition \ref{prop-X^o-B_R}, so is
%
\begin{math}
  \mathcal{R}_R^\natural : X^\odot\to X^\odot
                         : f\mapsto \chi_{B_R(0)}\, u ,
\end{math}
%
where
$u = \mathcal{K}\vert_{X^\odot} f$
and $\chi_{B_R(0)}$ denotes the characteristic function of
the open ball $B_R(0)\subset \mathbb{R}^N$.
Moreover, applying Lemma \ref{lem-X^o-infty}, we get
$\mathcal{R}_R^\natural \to \mathcal{K}\vert_{X^\odot}$
uniformly on $\overline{\mathcal{B}}_{X^\odot}$ as $R\to \infty$.
We invoke a well\--known approximation theorem
({Edwards} \cite[Theorem 9.2.6, p.~622]{Edwards} or
 {Yosida} \cite[Chapt.~X, Sect.~2, p.~278]{Yosida})
to conclude that also the limit operator
$\mathcal{K}\vert_{X^\odot} : X^\odot\to X^\odot$
must be compact.
\end{proof}


\section{Positivity and compactness for $q(x)$ nonradial}
\label{s:Proofs}

We begin with the proof of Theorem~\ref{thm-Positive-q(x)}
for potentials $q$ of class \eqref{class:Q(r)} only.

\subsection{Positivity for potentials of class \eqref{class:Q(r)}}
\label{ss:Positive-Q(r)}

Throughout this paragraph we consider only a radially symmetric potential
$q$ of class \eqref{class:Q(r)},
$q(x) = Q(|x|)$ for all $x\in \mathbb{R}^N$.
Therefore, all symbols
$\mathcal{A}$, $\Lambda$, $\varphi$, $X$, $X^\odot$, etc.
are considered only for this special type of potential.
We now prove Theorem~\ref{thm-Positive-q(x)}
for the Schr\"odinger equation
%
\begin{equation}
  - \Delta u + Q(|x|) u = \lambda u + f(x)
  \quad\mbox{in }\, X^\odot .
\label{Q(r):Schroed_lam}
\end{equation}
%

\begin{proposition}\label{prop-Positive-Q(r)}
Let  $Q(r)$ be of class~\eqref{class:Q(r)} and
$-\infty < \lambda < \Lambda$.
Assume that  $f\in X^\odot$ satisfies
$f\geq 0$ almost everywhere and
$f\not\equiv 0$ in~$\mathbb{R}^N$.
Then the (unique) solution $u\in X^\odot$ to
the Schr\"odinger equation \eqref{Q(r):Schroed_lam}
(in the sense of distributions on $\mathbb{R}^N$)
is given by
$u = (\mathcal{A} - \lambda I)^{-1}\vert_{X^\odot} f$
and satisfies
$u\geq c\varphi$ almost everywhere in $\mathbb{R}^N$,
with some constant $c\equiv c(f) > 0$.
%
\end{proposition}

This proposition is proved in
{Alziary} and {Tak\'a\v{c}}
\cite[Theorem 2.1, p.~284]{AlziaryTak}
under a slightly different growth hypothesis on
the monotone increasing potential $q(x)\equiv q(r)$.

\begin{proof}
The proof of our proposition follows the same pattern
as does the proof of Theorem 2.1 in \cite[pp.\ 289--290]{AlziaryTak}.
We leave the details to the reader.
\end{proof}

The results of the previous section ({\S}\ref{ss:compact-q_1<q_2})
allow us to finally remove
the restriction that $q$ be radially symmetric, i.e.,
we consider a potential $q: \mathbb{R}^N\to \mathbb{R}$ that satisfies
 \eqref{hyp:growth:q(x)}.

\subsection{Compactness of $\mathcal{K}\vert_{X}$ for $q$ nonradial}
\label{ss:Compact-q(x)}

\begin{proof}[Proof of  Theorem~\ref{thm-X-compact}]
{\rm Part~(a)}:
According to  \eqref{hyp:growth:q(x)},
potentials $q$, $Q_1$, and $Q_2$ satisfy \eqref{growth:q(x)}, that is,
%
\begin{equation*}
%\label{growth:q(x)}
  Q_1(|x|)\leq q(x)\leq Q_2(|x|)\leq C_{12}\, Q_1(|x|)
    \quad\mbox{for all } x\in \mathbb{R}^N .
\end{equation*}
%
Consequently, these potentials satisfy also
conditions \eqref{cond:q_0} in place of $q$.
We denote by $\Lambda_{q}$, $\Lambda_{Q_1}$, and $\Lambda_{Q_2}$
the principal eigenvalues of the Schr\"odinger operators
$\mathcal{A}_{q}$, $\mathcal{A}_{Q_1}$, and $\mathcal{A}_{Q_2}$
with potentials $q$, $Q_1$, and $Q_2$, respectively.
The associated eigenfunctions
$\varphi_{q}$, $\varphi_{Q_1}$, and $\varphi_{Q_2}$
are normalized by being positive throughout $\mathbb{R}^N$ and having
the $L^2(\mathbb{R}^N)$ norm ${}=1$.

First, we have
$0 < \Lambda_{Q_1}\leq \Lambda_q\leq \Lambda_{Q_2} < \infty$,
by Proposition~\ref{prop-q_1<q_2}, Part~(a).
From Proposition~\ref{prop-X-compact}
we infer that, given any $\lambda < \Lambda_{Q_1}$, the restriction
%
\begin{math}
  (\mathcal{A}_{Q_1} - \lambda I)^{-1}\vert_{X_{Q_1}}
  : X_{Q_1}\to X_{Q_1}
\end{math}
%
of the resolvent
$(\mathcal{A}_{Q_1} - \lambda I)^{-1}$
to $X_{Q_1}$ is compact (hence, also weakly compact).
By Proposition~\ref{prop-q_1<q_2}, Part~(c),
the same is true of the restrictions
$(\mathcal{A}_{q} - \lambda I)^{-1}\vert_{X_{Q_1}}$ and
$(\mathcal{A}_{Q_2} - \lambda I)^{-1}\vert_{X_{Q_1}}$
to $X_{Q_1}$.
Hence, we can apply
Corollary~\ref{cor-q_1<q_2} to conclude that
$\sup_{\mathbb{R}^N} (\varphi_{q} / \varphi_{Q_1}) < \infty$ and
$\sup_{\mathbb{R}^N} (\varphi_{Q_2} / \varphi_{Q_1}) < \infty$.
Equivalently, both
$X_{q}  \hookrightarrow X_{Q_1}$ and
$X_{Q_2}\hookrightarrow X_{Q_1}$ are continuous embeddings.

Second, let us denote
%
\begin{equation}
  V_j(r)\stackrel{{\rm def}}{=} Q_j(r) - \Lambda + \frac{(N-1)(N-3)}{4 r^2}
    \quad\mbox{for } r > r_0 ;\ j=1,2 ,
\label{def:V_j(r)}
\end{equation}
%
where $0 < r_0 < \infty$ is large enough, such that
$| {(N-1)(N-3)} | / {4 r_0^2} \leq 1$ and
$V_j(r) > 0$ for all $r > r_0$.
Consequently,
\[
  Q_j(r) - \Lambda - 1\leq V_j(r)\leq Q_j(r) - \Lambda + 1
    \quad\mbox{for all } r > r_0 ;\ j=1,2 .
\]
Therefore, we may apply first
Remarks \ref{rem-int-diff}, Part~{\rm (b)}, and \ref{rem-Titchmarsh},
and then Lemma \ref{lem-U1:U2} to conclude that
$\varphi_{Q_j}(r)$ is comparable with any positive solution $f_j(r)$ of
%
\begin{equation*}
  - f_j'' + Q_j(r) f_j = 0 \quad\mbox{for a.e. } r > r_0 ,
%\label{eq:f_j,f_j'}
\end{equation*}
%
such that $f_j(r)\to 0$ as $r\to \infty$.
The functions $f_1$ and $f_2$ are comparable by results from
paragraph {\S}\ref{ss:phi_1/phi_2},
Proposition \ref{lem-Titchmarsh} and Remark \ref{rem-Titchmarsh} for
$U_j = Q_j$; $j=1,2$.
Thus, we have obtained
%
\begin{equation}
\label{growth:phi_1/phi_2}
  0 < c'\leq \frac{\varphi_{Q_1}(r)}{\varphi_{Q_2}(r)}
        \leq c''< \infty
    \quad\mbox{for all } r > r_0 ,
\end{equation}
%
where $c'$ and $c''$ are some constants.
This formula yields
$\sup_{\mathbb{R}^N} (\varphi_{Q_1} / \varphi_{Q_2}) < \infty$ or, equivalently,
$X_{Q_1}\hookrightarrow X_{Q_2}$ is a continuous embedding.

Finally, let us rewrite the equation
$\mathcal{A}_{q} \varphi_{q} = \Lambda_{q} \varphi_{q}$
for
$\varphi_{q}\in X_{Q_1} = X_{Q_2}$ as
%
\begin{equation*}
    - \Delta\varphi_{q} + Q_2(|x|)\varphi_{q} = f(x)
  \quad\mbox{in }\, X_{Q_2}^\odot ,
%\label{Q_2(r):Schroed_0}
\end{equation*}
%
where
%
\begin{equation*}
  f(x) =
  \left[ Q_2(|x|) - q(x) + \Lambda_{q} \right] \varphi_{q}(x)
  \geq \Lambda_{q} \varphi_{q}(x) > 0 ,\quad x\in \mathbb{R}^N ,
\end{equation*}
%
by condition \eqref{growth:q(x)}.
Notice that
%
\begin{equation*}
  \sup_{\mathbb{R}^N} (\varphi_{q} / \varphi_{Q_1}) < \infty ,
    \quad
  \sup_{\mathbb{R}^N} (\varphi_{Q_2} / \varphi_{Q_1}) < \infty ,
    \quad\mbox{and}\quad
  \sup_{\mathbb{R}^N} (\varphi_{Q_1} / \varphi_{Q_2}) < \infty ,
\end{equation*}
%
combined with
$\int_{\mathbb{R}^N} Q_2\, \varphi_{Q_2}^2 \,{\rm d}x < \infty$,
yield
$\int_{\mathbb{R}^N} Q_2\, \varphi_{q}\, \varphi_{Q_2} \,{\rm d}x < \infty$,
that is,
%
\begin{math}
    Q_2\, \varphi_{q}\in X_{Q_2}^\odot
  = L^1(\mathbb{R}^N; \varphi_{Q_2} \,{\rm d}x) .
\end{math}
%
Consequently, also
$(Q_2 - q)\varphi_{q}\in X_{Q_2}^\odot$
which guarantees $f\in X_{Q_2}^\odot$.
We apply Proposition~\ref{prop-Positive-Q(r)} with $Q=Q_2$ and
$\lambda = 0 < \Lambda_Q$ to conclude that
$\inf_{\mathbb{R}^N} (\varphi_{q} / \varphi_{Q_2}) > 0$ or, equivalently,
$X_{Q_2}\hookrightarrow X_{q}$ is a continuous embedding.

Summarizing the results proved in this section for
$\varphi_{q}$, $\varphi_{Q_1}$, and $\varphi_{Q_2}$, we arrive at
$X_{q} = X_{Q_1} = X_{Q_2}$, i.e.,
%
\begin{math}
       \gamma_1\varphi_{q}
  \leq \varphi_{Q_1}, \varphi_{Q_2}
  \leq \gamma_2\varphi_{q}
\end{math}
%
everywhere in $\mathbb{R}^N$, where
$0 < \gamma_1\leq \gamma_2 < \infty$
are some constants.
As we already know that the restriction
$(\mathcal{A}_{q} - \lambda I)^{-1}\vert_{X_{Q_1}}$
to $X_{Q_1}$ is compact, Part~(a) follows immediately.

{\rm Part~(b)}:
$\;$
In the remaining part of the proof we abbreviate
$\mathcal{A} = \mathcal{A}_{q}$, $\Lambda = \Lambda_{q}$, and
$X = X_{q}$.
Let $\lambda\in \mathbb{C}$ be an eigenvalue of $\mathcal{A}$, that is,
$\mathcal{A} v = \lambda v$ for some $v\in L^2(\mathbb{R}^N)$, $v\neq 0$.
Since $\mathcal{A}$ is positive definite and selfadjoint on $L^2(\Omega)$,
its inverse $\mathcal{A}^{-1}$ is bounded on $L^2(\mathbb{R}^N)$.
Property \eqref{cond:q_0} implies that $\mathcal{A}^{-1}$ is also compact.
Consequently, $\lambda\in \mathbb{R}$ and $\lambda\geq \Lambda > 0$.
Given $v\in L^2(\mathbb{R}^N)$, $v\neq 0$, it follows that
equation $\mathcal{A} v = \lambda v$ is equivalent with
$\mathcal{A}^{-1} v = \lambda^{-1} v$.
By Part~(a), also the restriction
$\mathcal{A}^{-1}\vert_{X}$ to $X$ is compact.
Now we can apply Lemma~\ref{lem-Riesz-Thorin} with
$\mathcal{T} = \mathcal{A}^{-1}\vert_{X}$ compact on $X$
to obtain the conclusion of Part~(b).

{\rm Part~(c)}:
Assume that $\lambda\in \mathbb{C}$ is not an eigenvalue of $\mathcal{A}$.
With regard to Part~(a) we may restrict ourselves to the case
$\lambda\not\in (-\infty, \Lambda)$.
Hence, by the Riesz\--Schauder theory applied to $\mathcal{A}^{-1}$,
which is compact on $L^2(\mathbb{R}^N)$,
$\lambda$ is in the resolvent set of $\mathcal{A}$ and the resolvent
$\mathcal{K} = (\mathcal{A} - \lambda I)^{-1}$
is compact on $L^2(\mathbb{R}^N)$.
We refer to
{Edwards} \cite[Theorem 9.10.2, p.~679]{Edwards} or
{Yosida} \cite[Chapt.~X, Theorem 5.1, p.~283]{Yosida}
for the Riesz\--Schauder theory.
Consequently, the following identities hold on $L^2(\mathbb{R}^N)$:
%
\begin{equation}
    \mathcal{K} \left( \lambda^{-1} I - \mathcal{A}^{-1} \right)
  = \left( \lambda^{-1} I - \mathcal{A}^{-1} \right) \mathcal{K}
  = \lambda^{-1} \mathcal{A}^{-1}\, .
\label{res:(A-lam)^-1}
\end{equation}
%
In particular,
$\lambda^{-1}$ cannot be an eigenvalue of $\mathcal{A}^{-1}$.
So $\lambda^{-1}$ is not an eigenvalue of
$\mathcal{A}^{-1}\vert_{X}$ either.
The restriction
$\mathcal{A}^{-1}\vert_{X}$ being compact on $X$, by Part~(a),
we may apply Lemma~\ref{lem-Riesz-Thorin} with
$\mathcal{T} = \mathcal{A}^{-1}\vert_{X}$ again to conclude that
the restriction
$\lambda^{-1} I - \mathcal{A}^{-1}\vert_{X}$
of
$\lambda^{-1} I - \mathcal{A}^{-1}$ to $X$ has a bounded inverse, say,
%
\begin{math}
  \mathcal{L} =
  \left( \lambda^{-1} I - \mathcal{A}^{-1}\vert_{X} \right)^{-1}\, .
\end{math}
%
Hence, from \eqref{res:(A-lam)^-1} we deduce
%
\begin{math}
    \mathcal{K}\vert_{X}
  = \lambda^{-1} \mathcal{L} \left( \mathcal{A}^{-1}\vert_{X} \right)
\end{math}
%
which shows that also
$\mathcal{K}\vert_{X}$ is compact on $X$ as claimed.

The proof of {\rm Theorem~\ref{thm-X-compact}} is now complete.
\end{proof}


\subsection{Positivity for a nonradial potential $q(x)$}
\label{ss:Positive-q(x)}

\begin{proof}[Proof of Theorem~\ref{thm-Positive-q(x)}]
Let $-\infty < \lambda < \Lambda_{q}$ and
$u = (\mathcal{A}_{q} - \lambda I)^{-1}\vert_{X_{q}^\odot} f$.
Since $0\leq f\in X_{q}^\odot$,
we may apply the weak maximum principle
% (as in the proof of Proposition~\ref{prop-Positive-Q(r)})
to get $0\leq u\in X_{q}^\odot$.
Hence, it suffices to prove our theorem for
$g = \min\{ f,\, \varphi_{q}\}$ in place of $f$, that is, for
$0\leq f\leq \varphi_{q}$ a.e.\ and $f\not\equiv 0$ in $\mathbb{R}^N$.
This forces also
$0\leq u\leq (\Lambda_{q} - \lambda)^{-1} \varphi_{q}$ a.e.\ and
$u\not\equiv 0$ in $\mathbb{R}^N$,
by the weak maximum principle again.

Similarly as in
the proof of Theorem~\ref{thm-X-compact}, Part~(a) above,
let us rewrite the equation
$\mathcal{A}_{q} u = \lambda u + f$
for $u\in X_{q}$, with $f\in X_{q}$, $X_{q} = X_{Q_1} = X_{Q_2}$, as
%
\begin{equation*}
    - \Delta u + Q_2(|x|) u = \lambda u + g(x)
  \quad\mbox{in }\, X_{Q_2}^\odot ,
%\label{Q_2(r):Schroed_lam}
\end{equation*}
%
where
%
\begin{equation*}
  g(x) =
  \left[ Q_2(|x|) - q(x) \right] u(x) + f(x)
  \geq f(x) ,\quad x\in \mathbb{R}^N ,
\end{equation*}
%
by condition \eqref{growth:q(x)} and
$u\geq 0$ a.e.\ in $\mathbb{R}^N$.
Again, we combine Corollary~\ref{cor-X-compact} with
$$
\int_{\mathbb{R}^N} Q_2\, \varphi_{Q_2}^2 \,{\rm d}x < \infty
$$
to get
$\int_{\mathbb{R}^N} Q_2\, u\, \varphi_{Q_2} \,{\rm d}x < \infty$,
that is,
%
\begin{math}
    Q_2\, u\in X_{Q_2}^\odot
  =\break L^1(\mathbb{R}^N; \varphi_{Q_2} \,{\rm d}x) .
\end{math}
%
Consequently, also
$(Q_2 - q) u\in X_{Q_2}^\odot$
which guarantees $g\in X_{Q_2}^\odot$.
We apply Proposition~\ref{prop-Positive-Q(r)} with $Q=Q_2$ and
$\lambda < \Lambda_{q}\leq \Lambda_Q = \Lambda_{Q_2}$
to conclude that
$\inf_{\mathbb{R}^N} (u / \varphi_{Q_2}) > 0$ or, equivalently,
$u\geq c\varphi_{q}$ a.e.\ in $\mathbb{R}^N$,
with some constant $c\equiv c(f) > 0$.
\end{proof}


\section{Appendix} \label{s:Appendix}

\subsection{An example of a monotone radial potential}
\label{ss:Examples}

Here we give an example of a radially symmetric potential
$q(x) = q(r)$ which belongs to class \eqref{class:Q(r)},
but does {\it not } belong to the analogue of this class defined in
{Alziary}, {Fleckinger}, and {Tak\'a\v{c}}
\cite[eqs.\ (13) and (14)]{AFT-3}.
More precisely, it fails to satisfy condition \eqref{e:int-deriv}
for {\it any } $\gamma > 0$
(\cite[eq.~(14)]{AFT-3}).
This example illustrates how ``large'' class \eqref{class:Q(r)}
actually is.

For $r\in \mathbb{R}_+$ we set
either $q'(r) = 0$ or else $q'(r) = 2\, q(r)^{3/2}$,
which yields a ``very fast'' growth of $q(r)$ on a sequence of
pairwise disjoint, nonempty intervals
$(n - \varrho_n, n + \varrho_n)$; $n=1,2,3,\dots$,
of total length
$2\sum_{n=1}^\infty \varrho_n = 1$, where
$\varrho_n\to 0$ sufficiently fast as $n\to \infty$, say,
$\varrho_n = \mathcal{O}(1/n^3)$.

\begin{example}[{\cite[Example 3.6]{AFT-3}}] \label{exam-q(r):mono} \rm
We define $q: \mathbb{R}_+\to (0,\infty)$ by
$q(r) = \theta(r)^{-2}$ for $r\in \mathbb{R}_+$, where
$\theta: \mathbb{R}_+\to (0,1]$ is
a monotone decreasing, piecewise linear, continuous function
defined as follows:
Let
$\{ \varrho_n\}_{n=1}^\infty \subset (0,1/2)$
be a sequence of numbers satisfying
%
\begin{equation}
 \sum_{n=1}^\infty \varrho_n = {1}/{2} \,.
\label{sum:rho_n=1/2}
\end{equation}
%
Given $r\geq 0$, we set $\theta(0) = 1$ and
%
\begin{equation*}
%\label{e:deriv=-1/0}
  \frac{\mathrm{d}\theta}{\mathrm{d}r}(r) =
  \begin{cases}
  -1 & \mbox{if $|r-n| < \varrho_n$ for some $n\in \mathbb{N}$;} \\
   0 & \mbox{otherwise,} \\
  \end{cases}
\end{equation*}
%
where $\mathbb{N} = \{ 1,2,3,\dots\}$.
Setting $R_0 = 1$ and abbreviating
%
\[
  R_n = 1 - 2\sum_{k=1}^n \varrho_k > 0
  \quad\mbox{for } n=1,2,\dots ,
%\label{sum:R_n}
\]
%
we compute for $r\geq 0$:
%
\begin{equation*}
  \theta(r) =
\begin{cases}
  1 & \mbox{if } 0\leq r\leq 1 - \varrho_1 ; \\
R_{n-1} - ((r-n) + \varrho_n)
   & \mbox{if $|r-n| < \varrho_n$ for some $n\in \mathbb{N}$;} \\
R_n &\mbox{if } \varrho_n\leq r-n\leq 1 - \varrho_{n+1}
       \mbox{ for some $n\in \mathbb{N}$.}
\end{cases}
\end{equation*}
%
Clearly,
$\theta: \mathbb{R}_+\to (0,1]$ is
monotone decreasing, piecewise linear, and continuous.
It satisfies $\theta(r)\to 0$ as $r\to 0+$, by \eqref{sum:rho_n=1/2}.
Next, we compute
%
\begin{equation*}
%\label{sum:int-Q^-1/2}
\begin{aligned}
    \int_{1 - \varrho_1}^{\infty} \theta(r) \,\mathrm{d}r
& = \sum_{n=1}^\infty (R_{n-1} - \varrho_n) \cdot 2\varrho_n
  + \sum_{n=1}^\infty R_n (1 - \varrho_{n+1} - \varrho_n)
\\
&  < 1 + 2\sum_{n=1}^\infty R_n \,.
\end{aligned}
\end{equation*}
%
Furthermore, for any $\gamma > 0$ we get
%
\begin{equation*}
%\label{sum:int-deriv}
\begin{aligned}
& \int_{0}^{\infty}
  \left|    \frac{{\rm d}}{{\rm d}r} \left( q(r)^{-1/2} \right)
  \right|^{\gamma} q(r)^{1/2} \,{\rm d}r \\
& =  \sum_{n=1}^\infty \int_{|r-n| < \varrho_n}
    \left[ R_{n-1} - ((r-n) + \varrho_n) \right]^{-1} \,\mathrm{d}r
  > 2\sum_{n=1}^\infty \varrho_n\, R_{n-1}^{-1} \,.
\end{aligned}
\end{equation*}
%

We will have an example of a potential $q(r)$ with the desired properties
as soon as we find a sequence
$\{ \varrho_n\}_{n=1}^\infty \subset (0,1/2)$
that satisfies the following conditions:
%
\begin{equation}
 \sum_{n=1}^\infty \varrho_n = {1}/{2} \,,\quad
%\label{sum:rho_n=1/2}
  \sum_{n=1}^\infty R_n < \infty \,,\quad\mbox{and }\quad
%\label{sum:R_n<infty}
  \sum_{n=1}^\infty \varrho_n\, R_{n-1}^{-1} = \infty \,.
\label{sum:rho_n/R_n=infty}
\end{equation}
%
A simple choice of such $\varrho_n$'s is, for instance,
%
\begin{equation*}
    \varrho_n
  = \frac{1}{2}\Big( \frac{1}{n^2} - \frac{1}{(n+1)^2} \Big)
  = \frac{1 + \frac{1}{2n}}{n (n+1)^2}
    \quad\mbox{for } n=1,2,\dots \,,
%\label{def:rho_n}
\end{equation*}
%
which renders
%
\begin{equation*}
  R_n = (n+1)^{-2} \quad\mbox{and}\quad
    \varrho_n\, R_{n-1}^{-1}
  = \frac{1 + \frac{1}{2n}}{ n( 1 + \frac1n)^2 }
    \quad\mbox{for } n=1,2,\dots \,.
\end{equation*}
%
It is easy to see that these $\varrho_n$'s satisfy all conditions in
\eqref{sum:rho_n/R_n=infty}.
\end{example}
%

\subsection{Potentials with at most (exponential) power growth}
\label{ss:power-grow}

Here we verify the claims from cases {\rm (i)} and {\rm (ii)}
in Remark \ref{rem-int-diff}, Part~{\rm (a)}.
Let us set
$S(r) = Q_1(r)^{1/2} + Q_2(r)^{1/2}$ for $r\geq r_0$.
Motivated by the computations in \eqref{case:int-diff},
we simply compare the end point value $S(s)$ with the average value
%
\begin{math}
  (s-r)^{-1} \int_r^s S(t) \,{\rm d}t
\end{math}
%
for all $r_0\leq r < s < \infty$.

\subsubsection*{Potentials with at most power growth}

\begin{lemma}\label{lem-power-grow}
Assume that
$S(t) = Q_1(t)^{1/2} + Q_2(t)^{1/2}$ ($t\geq r_0$) satisfies
%
\begin{equation}
  S(s)\leq \frac{C}{s-r} \int_r^s S(t) \,\mathrm{d}t
  \quad\mbox{for all }\; r_0\leq r < s < \infty ,
\label{e:power-growth}
\end{equation}
%
where $C\geq 1$ is a constant.
If condition \eqref{equiv:int-diff} is valid then also
condition \eqref{e:int-diff} is satisfied.
In particular, given any constant $\alpha\geq 0$, the function
$S(t) = \mathrm{const}\cdot t^{\alpha}\geq 0$
obeys inequality \eqref{e:power-growth}
with $C = 1+\alpha$.
%
\end{lemma}

\begin{proof}
From \eqref{e:power-growth} we deduce
%
\begin{equation*}
\begin{split}
  \int_{r_0}^s \exp\Big( - \int_r^s S(t) \,\mathrm{d}t \Big)
    \,\mathrm{d}r
&\leq \int_{r_0}^s \exp\Big( - C^{-1} (s-r) S(s) \Big)
    \,\mathrm{d}r\\
&= \frac{C}{S(s)}
    \big[ 1 - \exp\big( - C^{-1} (s-r_0) S(s) \big) \big] \\
&\leq \frac{C}{S(s)}
\end{split}
\end{equation*}
%
for all $s\geq r_0$.
Consequently, \eqref{e:int-diff} follows from
%
\begin{equation*}
\begin{aligned}
& \int_{r_0}^\infty ( Q_2(s) - Q_1(s) ) \int_{r_0}^s \exp
    \Big( - \int_r^s [ Q_1(t)^{1/2} + Q_2(t)^{1/2} ] \,\mathrm{d}t
    \Big)
  \,\mathrm{d}r \,\mathrm{d}s \\
& \leq {} \int_{r_0}^\infty ( Q_2(s) - Q_1(s) ) \,\frac{C}{S(s)}\,
       \,\mathrm{d}s \\
&= C\int_{r_0}^\infty
    \left( Q_2(s)^{1/2} - Q_1(s)^{1/2} \right) \,\mathrm{d}s
  < \infty \,,
\end{aligned}
\end{equation*}
%
by condition \eqref{equiv:int-diff}.

If $S(t) = c\cdot t^{\alpha}$,
with some constants $c, \alpha\geq 0$, then we have
%
\begin{equation*}
\begin{split}
(s-r) S(s) &= c (s-r) s^{\alpha}
   = c\, s^{1+\alpha} (1 - (r/s)) \\
  & \leq c\, s^{1+\alpha} \left( 1 - (r/s)^{1+\alpha} \right)\\
& = c ( s^{1+\alpha} - r^{1+\alpha} ) \\
& = c (1+\alpha) \int_r^s t^{\alpha} \,\mathrm{d}t \\
& = (1+\alpha) \int_r^s S(t) \,\mathrm{d}t
\end{split}
\end{equation*}
%
for all $r_0\leq r\leq s < \infty$.
\end{proof}

\subsubsection*{Potentials with at most exponential power growth}
Now we weaken condition \eqref{e:power-growth} as follows.


\begin{lemma}\label{lem-exp-power-gr}
Assume that
$S(t) = Q_1(t)^{1/2} + Q_2(t)^{1/2}$ ($t\geq r_0$) satisfies
%
\begin{equation}
       \frac{S(s)}{1 + \log^+ S(s)}
  \leq \frac{C}{s-r} \int_r^s S(t) \,{\rm d}t
  \quad\mbox{for all } r_0\leq r < s < \infty ,
\label{e:exp-power-gr}
\end{equation}
%
where $\log^+ = \max\{ \log, 0\}$ and  $C\geq 1$ is a constant.
Then condition \eqref{exp:int-diff} implies that also
condition \eqref{e:int-diff} holds.
In particular, given any constants
$\alpha, \beta\geq 0$ and  $\gamma > 0$, the function
$S(t) = \gamma\cdot \exp( \beta t^{\alpha} ) > 0$
obeys inequality  \eqref{e:exp-power-gr} with
$C = \alpha_1 ( 1 + \log^+ \gamma^{-1} )$
where
$\alpha_1 = \max\{ \alpha, 1\}$.
%
\end{lemma}

\begin{proof}
From \eqref{e:exp-power-gr} we deduce
%
\begin{equation*}
\begin{split}
 \int_{r_0}^s \exp\Big( - \int_r^s S(t) \,\mathrm{d}t \Big)
    \,\mathrm{d}r
&\leq \int_{r_0}^s
    \exp\Big(  - \,\frac{(s-r) S(s)}{ C\, (1 + \log^+ S(s)) }\Big)
    \,\mathrm{d}r
\\
&  = \frac{ C\, (1 + \log^+ S(s)) }{S(s)}
    \Big[ 1 -     \exp\Big( - \,\frac{(s-r) S(s)}{ C\, (1 + \log^+ S(s)) }
\Big)     \Big]
\\
& \leq \frac{ C\, (1 + \log^+ S(s)) }{S(s)}
\end{split}
\end{equation*}
%
for all $s\geq r_0$.
Consequently, \eqref{e:int-diff} follows from
%
\begin{equation*}
\begin{aligned}
& \int_{r_0}^\infty ( Q_2(s) - Q_1(s) ) \int_{r_0}^s \exp
    \Big( - \int_r^s [ Q_1(t)^{1/2} + Q_2(t)^{1/2} ] \,\mathrm{d}t
    \Big)
  \,\mathrm{d}r \,\mathrm{d}s
\\
&  \leq \int_{r_0}^\infty
    ( Q_2(s) - Q_1(s) ) \,\frac{ C\, (1 + \log^+ S(s)) }{S(s)}\,
      \,\mathrm{d}s
\\
& = C\int_{r_0}^\infty
    \left( Q_2(s)^{1/2} - Q_1(s)^{1/2} \right)
       \left[ 1 + \log^+ ( Q_1(r)^{1/2} + Q_2(r)^{1/2} ) \right]
    \,\mathrm{d}r
  < \infty \,,
\end{aligned}
\end{equation*}
%
by condition \eqref{exp:int-diff}.

We leave the example
$S(t) = \gamma\cdot \exp( \beta t^{\alpha} )$
to the reader as an easy exercise.
\end{proof}

\subsection{Extensions of certain symmetric operators}
\label{ss:Exten-oper}

We present a few obvious, but necessary facts
about extensions of bounded symmetric operators defined on $X$ to
$L^2(\mathbb{R}^N)$ and $X^\odot$.
The following lemma is an easy consequence of
the Riesz\--Thorin interpolation theorem
({M.~Reed} and {B.~Simon}
 \cite[Sect.\ IX.4, Theorem IX.17, p.~27]{RSimon-II}).
It is applied to the resolvent
$\mathcal{K} = (\mathcal{A} - \lambda I)^{-1}$ on $L^2(\mathbb{R}^N)$,
for $\lambda < \Lambda$,
which is bounded on $X$ by inequality \eqref{e:max-princ_X},
and to similar operators as well.

\begin{lemma}\label{lem-Riesz-Thorin}
Let  $q$, $\varphi$, $X$, and  $X^\odot$ be as in
{\rm Section~\ref{s:Main}}.
Assume that
$\mathcal{T}: X\to X$ is a bounded linear operator
that satisfies the symmetry condition
%
\begin{equation}
   \int_{\mathbb{R}^N} (\mathcal{T} f)\, \bar{g} \,{\rm d}x
  = \int_{\mathbb{R}^N} f\, \overline{(\mathcal{T} g)} \,{\rm d}x
  \quad\mbox{for all } f,g\in X .
\label{symm:T_X}
\end{equation}
%
Then $\mathcal{T}$ possesses a unique extension
$\mathcal{T}\vert_{X^\odot}$
to a bounded linear operator on $X^\odot$,
$\mathcal{T}$ is the adjoint of  $\mathcal{T}\vert_{X^\odot}$, and
$\mathcal{T}\vert_{X^\odot}$
restricts to a bounded selfadjoint operator
$\mathcal{T}\vert_{L^2(\mathbb{R}^N)}$ on $L^2(\mathbb{R}^N)$.
Moreover, the operator norms of
$\mathcal{T}\vert_{L^2(\mathbb{R}^N)}$, $\mathcal{T}\vert_{X^\odot}$, and
$\mathcal{T}$, respectively, satisfy
%
\begin{equation}
  \| \mathcal{T}\vert_{L^2(\mathbb{R}^N)} \|_{ L^2(\mathbb{R}^N)\to L^2(\mathbb{R}^N) }
  \leq \| \mathcal{T}\vert_{X} \|_{X^\odot\to X^\odot}
  =    \| \mathcal{T} \|_{X\to X} < \infty \,.
\label{ineq-Riesz-Thorin}
\end{equation}
%
The spectrum of  $\mathcal{T}\vert_{L^2(\mathbb{R}^N)}$
is contained in the spectrum of  $\mathcal{T}$.
Finally, if  $\mathcal{T}$ is compact, then so is
$\mathcal{T}\vert_{L^2(\mathbb{R}^N)}$ and their spectra coincide;
in particular, if
$\mathcal{T}\vert_{L^2(\mathbb{R}^N)} v = \lambda v$ for some
$\lambda\in \mathbb{C}\setminus \{ 0\}$ and
$v\in L^2(\mathbb{R}^N)\setminus \{ 0\}$, then
$\lambda\in \mathbb{R}$ and  $v\in X$.
%
\end{lemma}

This lemma is proved in {Alziary}, {Fleckinger}, and {Tak\'a\v{c}}
\cite[Lemma 4.3]{AFT-3}.

\subsection*{Acknowledgment}

The authors would like to express their sincere thanks to
{\it Professor Jacqueline Fleckinger-Pell\'e}
(Universit\'e Toulouse~1, France)
for her tremendous support of their joint work on this and
a number of other papers about Schr\"odinger operators.
Since 1993 both authors have enjoyed working with her and obtaining
numerous interesting scientific results.
During the pastime they have appreciated a lasting friendship with her,
her readiness to help, and her everlasting optimism that
she has always been willing to share with many others.

This work of all three authors was supported in part by
        le Minist\`ere des Affaires \'Etrang\`eres (France)
        and
        the German Academic Exchange Service (DAAD, Germany)
        within the exchange program ``PROCOPE''.
        A part of this research was performed when
        P.~Tak\'a\v{c} was a visiting professor at
        CEREMATH -- UMR MIP,
        Universit\'e Toulouse~1, Toulouse, France.


\begin{thebibliography}{99}

\bibitem{AFT-1}
B.~Alziary, J.~Fleckinger and P.~Tak\'a\v{c};
\emph{An extension of maximum and anti-maximum principles
     to a Schr\"odinger equation in $\mathbb{R}^2$},
J.~Differential Equations, {\bf 156} (1999), 122--152.

\bibitem{AFT-2}
B.~Alziary, J.~Fleckinger and P.~Tak\'a\v{c};
\emph{Positivity and negativity of solutions to
     a Schr\"o\-din\-ger equation in $\mathbb{R}^N$},
Positivity, {\bf 5}(4) (2001), 359--382.

\bibitem{AFT-3}
B.~Alziary, J.~Fleckinger and P.~Tak\'a\v{c};
\emph{Ground\--state positivity, negativity, and compactness for
     a Schr\"odinger operator in $\mathbb{R}^N$},
J.~Funct. Anal., {\bf 245} (2007), 213--248.
{\it Online}: doi: 10.1016/j.jfa.2006.12.007.

\bibitem{AlziaryTak}
B.~Alziary and P.~Tak\'a\v{c};
\emph{A pointwise lower bound for positive solutions
     of a Schr\"odinger equation in $\mathbb{R}^N$},
J.~Differential Equations, {\bf 133}(2) (1997), 280--295.

\bibitem{ClemPel}
Ph.~Cl\'ement and L.~A. Peletier;
\emph{An anti-maximum principle for second order elliptic operators},
J.~Differential Equations, {\bf 34} (1979), 218--229.

\bibitem{Davies}
E.~B. Davies;
{\sl ``Heat Kernels and Spectral Theory''},
Cambridge Univ.\ Press, Cambridge, 1989.

\bibitem{DavSimon}
E.~B. Davies and B.~Simon;
\emph{Ultra\-contractivity and the heat kernel for
     Schr\"odinger operators and Dirichlet Laplacians},
J.~Funct. Anal., {\bf 59} (1984), 335--395.

\bibitem{Edwards}
R.~E. Edwards;
{\sl ``Functional Analysis: Theory and Applications''},
Holt, Rinehart and Winston, Inc., New York, 1965.

\bibitem{EdmEvans}
D.~E. Edmunds and W.~D. Evans;
{\sl ``Spectral Theory and Differential Operators''},
Oxford University Press, Oxford, 1987.

\bibitem{LCEvans}
L.~C. Evans;
{\sl ``Partial Differential Equations''},
Graduate Studies in Mathematics, Vol.~{\bf 19},
American Math.\ Society, Providence, R.~I., 1998.

\bibitem{GilbargTrud}
D.~Gilbarg and N.~S. Trudinger;
{\sl ``Elliptic Partial Differential Equations of Second Order''},
Springer-Verlag, New York--Berlin--Heidelberg, 1977.

\bibitem{Hartman}
Ph.~Hartman;
{\sl ``Ordinary Differential Equations''}, 2nd Ed.,
Birkh\"auser, Boston--Basel--Stuttgart, 1982.

\bibitem{Hartman-Wint}
Ph.~Hartman and A.~Wintner;
\emph{Asymptotic integrations of linear differential equations},
American J. Math., {\bf 77}(1) (1955), 45--86.

\bibitem{HoffSwetina}
M.~Hoffmann-Ostenhof, T.~Hoffmann-Ostenhof, and J.~Swetina,
\emph{Pointwise bounds on the asymptotics of
     spherically averaged $L^2$-solutions of
     one-body Schr\"odinger equations},
Ann. Inst. Henri Poincar\'e Anal. Physique th\'eorique,
{\bf 42}(4) (1985), 341--361.

\bibitem{Hoffmann}
M.~Hoffmann-Ostenhof;
\emph{On the asymptotic decay of $L^2$-solutions of
     one-body Schr{\"o}\-din\-ger equations in unbounded domains},
Proc. Roy. Soc. Edinburgh, {\bf 115A} (1990), 65--86.

\bibitem{Kato}
T.~Kato;
{\sl ``Perturbation Theory for Linear Operators''},
in \emph{Grundlehren der ma\-the\-ma\-ti\-schen Wissenschaften},
Vol.~{\bf 132}.
Springer\--Verlag, New York\--Berlin\--Hei\-del\-berg, 1980.

\bibitem{ProtterWein}
M.~H. Protter and H.~F. Weinberger;
{\sl ``Maximum Principles in Differential Equations''},
Springer-Verlag, New York--Berlin--Heidelberg, 1984.

\bibitem{RSimon-II}
M.~Reed and B.~Simon;
{\sl ``Methods of Modern Mathematical Physics, Vol.~II:
       Fourier Analysis, Self-Adjointness''},
Academic Press, Inc., Boston, 1975.

\bibitem{RSimon-IV}
M.~Reed and B.~Simon;
{\sl ``Methods of Modern Mathematical Physics, Vol.~IV:
       Analysis of Operators''},
Academic Press, Inc., Boston, 1978.

\bibitem{Sweers}
G.~Sweers;
\emph{Strong positivity in $C(\overline{\Omega})$ for elliptic systems},
Math. Z., {\bf 209} (1992), 251--271.

\bibitem{Takac}
P.~Tak\'a\v{c};
\emph{An abstract form of maximum and anti-maximum principles of
     Hopf's type},
J. Math. Anal. Appl., {\bf 201} (1996), 339--364.

\bibitem{Titchmarsh}
E.~C. Titchmarsh;
{\sl ``Eigenfunction Expansions Associated with
     Second\--order Differential Equations''}, Part I,
Oxford University Press, Oxford, England, 1962.

\bibitem{Yosida}
K.~Yosida;
{\sl ``Functional Analysis''}, 6th Ed.,
in \emph{Grundlehren der ma\-the\-ma\-ti\-schen Wissenschaften},
Vol.~{\bf 123}.
Springer\--Verlag, New York\--Berlin\--Hei\-del\-berg, 1980.

\end{thebibliography}

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